Two early church writings can be seen to address this issue: the Didache and Justin Martyr's First Apology.
Early evidence for a trinitarian formula in baptism might be found in the Didache, which is commonly dated to the late first century:
And concerning baptism, baptize this way: Having first said all these things, baptize into the name of the Father, and of the Son, and of the Holy Spirit, in living water. But if you have not living water, baptize into other water; and if you can not in cold, in warm. But if you have not either, pour out water thrice upon the head into the name of Father and Son and Holy Spirit. [Chapter 7]
The author of this text includes the language of Matthew 28:19 twice alongside more specific instructions regarding the method of baptism. Of course, because the language is the same as that of Matthew, the same argument could be applied here: that the text really is referring to the singular "name" of Jesus. Interestingly, however, despite the very explicit instructions contained here regarding the mode of baptism, there is no explicit teaching here that the "name" of the Father, Son, and Holy Spirit is actually Jesus: instead, the phrase is merely repeated twice.
Thus the Didache contains perhaps a stronger statement for a trinitarian formula than Matthew 28:19, given the context, but the instruction here is still inconclusive.
Sometime between AD 147 and 161, Justin Martyr wrote his First Apology, and in it we have an early clear expression of the trinitarian formula being used in baptism. He writes:
For, in the name of God, the Father and Lord of the universe, and of our Saviour Jesus Christ, and of the Holy Spirit, they then receive the washing with water. [Chapter 61]
Notice here that Justin specifically references "Jesus Christ" instead of the "Son." Justin thus goes against the "Jesus-only" interpretation, since his words would reduce their interpretation to a tautology, "the name of Jesus is Jesus." Later in the same chapter, he makes this understanding even more clear, by referring to the singular name of the Father, name of Jesus, and name of the Holy Spirit:
There is pronounced over him who chooses to be born again, and has repented of his sins, the name of God the Father and Lord of the universe [...] And in the name of Jesus Christ, [...] and in the name of the Holy Ghost.
This is by no means the last word on this subject, as it was a matter of significant debate in the early church. But at the very least we have explicit use of the trinitarian (three-part) formula dating to the middle of the second century, and some inconclusive evidence for it dating to the first.