It's at least true that others in the early church taught doctrines that resembled Origen's. However, not all who did would normally be considered "church fathers," and even among those who unambiguously are, there's debate over how similar their teachings actually were to Origen's.
Of course, proponents of the doctrine will naturally find more similarities than others, as will be seen shortly. However, even a "mainstream" treatment (Philip Schaff, History of the Christian Church) mentions others as holding views similar to Origen:
Even Gregory of Nyssa, although, like Nemesius and Cyril of Alexandria, he supposed the soul to be created before the body, compares Origen's theory to the heathen myths and fables. (Vol. III., page 831)
Schaff also mentions on the same page that Augustine spoke at least relatively favorably of the doctrine in De libero arbitrio, though he later emphatically rejected it.
Turning now to proponents of the doctrine, whose objectivity some might call into question, we find many claims of support from the church fathers. First, Henry More (1614–1687) wrote a number of works on the subject, and tracks the history of the doctrine in the preface to his Collection of Several Philosophical Writings. Starting on page xx, he cites Clement of Alexandria (Stromata, Books I and III, and a letter to Julius Cassianus), Basil of Caesarea, Gregory of Nazianzus, Synesius, Arnobius, Prudentius, and Augustine (though he cites only De libero arbitrio, and not Augustine's later works). Unfortunately, most of More's specific examples are in Latin or Greek, making it difficult for non-scholars to judge the original intent of the quoted authors. Those who want to examine the quality of his interpretations can review them in the linked work.
After the original generations of Christians, we find the early Church Fathers, such as Justin Martyr (AD 100–l65), St. Clement of Alexandria (AD 150–220), and Origen (AD 185–254) teaching the pre-existence of souls, taking up reincarnation or one or another aspect of reimbodiment. Examples are scattered through Origen's works, especially Contra Celsum (1, xxxii), where he asks: "Is it not rational that souls should be introduced into bodies, in accordance with their merits and previous deeds . . . ?" And in De Principiis he says that "the soul has neither beginning nor end." St. Jerome (AD 340–420), translator of the Latin version of the Bible known as the Vulgate, in his Letter to Demetrias (a Roman matron), states that some Christian sects in his day taught a form of reincarnation as an esoteric doctrine, imparting it to a few "as a traditional truth which was not to be divulged."
Synesius (AD 370–480), Bishop of Ptolemais, also taught the concept, and in a prayer that has survived, he says: "Father, grant that my soul may merge into the light, and be no more thrust back into the illusion of earth." Others of his Hymns, such as number III, contain lines clearly stating his views, and also pleas that he may be so purified that rebirth on earth will no longer be necessary. In a thesis on dreams, Synesius writes: "It is possible by labor and time, and a transition into other lives, for the imaginative soul to emerge from this dark abode." This passage reminds us of verses in the Revelation of John (3:12), with its symbolic, initiatory language leading into: "Him that overcometh will I make a pillar in the temple of my God, and he shall go no more out."
Origen did teach about the pre-existence of souls as it was widely held during his time among many respected Church fathers.
Origen quoted Romans 9:11-14 as evidence for his position regarding the pre-existence of souls:
11 (For the children being not yet born, neither having done any good or evil, that the purpose of God according to election might stand, not of works, but of him that calleth;) 12 It was said unto her, The elder shall serve the younger. 13 As it is written, Jacob have I loved, but Esau have I hated. 14 What shall we say then? Is there unrighteousness with God? God forbid.
Origen argued that God could not love Jacob and hate Esau until Jacob had done something worthy of love and Esau had done something worthy of hatred; so then, these verses can only mean that Jacob and Esau had not yet done good or evil In This Life and their conduct Before This Life was the reason why Esau would serve Jacob. Origen rejected the position that God loves or hates a soul based on its inclination toward good or evil, and before the soul actually commits a good or evil act. God, being the perfect creator of all souls and their inclinations, knows perfectly well each soul's inclination toward good or evil.
"Before I formed you in the womb I knew you, And before you were born I consecrated you; I have appointed you a prophet to the nations."
Clearly, God is stating that Jeremiah was selected before he was formed in the womb and before he was born for God's special purpose.
Origen's writings did not support reincarnation. Origen's writings reflected a belief in Christian metempsychosis. The difference is clear if anyone studies it. Reincarnation is the belief that there is an endless automatic recycling of souls into human or animal forms. Origin, on the other hand, taught that human souls only incarnate in human bodies; it was not an automatic event, but determined by the will of God.
Many Christians confuse Origen's uncorrupted and original teachings with the later developments by followers in different geographical locations that became known as "origenism." Origenism is a corruption of Origen's original teachings based upon ignorance and gross misunderstandings as well as slander, gossip, and judgments by those who ignorantly misunderstood what Origen was trying to convey. It was "origenism" that was anathematized and NOT Origen himself. Origen was already dead and in Heaven with God by the time "origenism" was anathematized.
It is common among for people to confuse reincarnation with Origin's unique teachings on Christian metempsychosis which also was distinctly different from Plato's metempsychosis; likewise, it is also common to confuse the followers of Origen and their ignorant corruption of his teachings with Origen himself and his original teachings.
I haven't read everything cited in the long answer above, but I have read Justin Martyr and it isn't even discussed. The prayer cited at the end is clearly not about reincarnation, but about the states of mind and spirit that the individual in this life may go through, coming into light and falling into darkness, discussed by many regarding THIS life only and the possible repurcussions at the last judgement.
Jeremiah's cite above doesn't relate to preexistence - before I formed you doesn't have to mean before incarnation while you existed, but forming after conception, or it could mean the intention for Jeremiah to come into existence and plans for him.
I think a lot of confusion exists relating to the idea that the soul is created separately from but at the same time as the physical conception, and added to it instantly, all of this in a matter of seconds if not less,
Traducianism states that the soul material (for lack of a better term) is passed to the child from the parents like the physical material is, and at the same time. The soul material is what makes the physical material alive. Paul takes this for granted in explaining how Jesus' Melchizedec priesthood (drawing on a statement in Psalms) is superior to that of Aaron and Levi and not derived from them.
Also Adam became (not acquired) a living soul, animals also have the breath of life so are living souls as stated in the part of Genesis describing what died in the Flood, just different kinds of souls from us, and Ecclesiastes refers to the animal spirit speculating on where it goes at death.