First of all, there is no biblical basis for a spirit in a natural person who is ungodly and unsaved unlike someone saved by the perfect and complete work of Jesus Christ on the cross.
Soul alone or with Spirit?
Unsaved person: body and soul only but no Spirit. Soul is a also known as spirit of life, hence man in Genesis is described as flesh wherein is the spirit of life. So context is king to understand the word spirit whenever it occurs. When we read about Pharaoh's spirit was troubled here is soul, spirit of life in the flesh of Pharaoh (Gen 41:8).
Saved person: body, soul, then dwelling by the Holy Ghost (Holy Spirit) in the new birth (new creation). Make also the distinction between the Holy Ghost dwelling in a person (always saved) and the Holy Ghost coming upon a person (could be saved but not necessarily as Saul of the OT).
Jesus Christ is an exception as He has eternal Spirit (Heb 9:14) referring to His deity. He made his Soul naked for death, he also bare our sins in His own Body.
Keep in mind, in earlier Scriptures, we read about flesh and blood, blood is often linked with the soul there, so flesh and blood means body and soul.
And the LORD God formed man [of] the dust of the ground, and breathed
into his nostrils the breath of life; and man became a living soul.
As you can see no Spirit but breath of life (a special word in Hebrew was used here to distinguish man from animals), but not the Spirit here definitely, as the natural man has no Spirit. There is no evidence that man was tripartite at first, he becomes so once born from above by the Spirit.
That which is born of the flesh is flesh; and that which is born of
the Spirit is spirit. (John 3:6)
So your question seems to be, given the above, as if directed toward the saved souls with a spirit but not toward unsaved ones who have no spirit but souls. Either way, there is consciousness.
Evidence of consciousness after death
The evidence of that soul being conscious after death is in the parable of the rich man and Lazarus the beggar in Luke 16:
And he cried and said, Father Abraham, have mercy on me, and send
Lazarus, that he may dip the tip of his finger in water, and cool my
tongue; for I am tormented. But Abraham
said, Son, remember that thou in thy lifetime receivedst thy good
things, and likewise Lazarus evil things: but now he is comforted, and
thou art tormented. (Luke 16:24,25)
Obviously the rich (unsaved) was in pain, had he not been conscious how could he state his torment then, in contrast, Lazarus (saved) was comforted.
This witness was from the NT, we have something about Samuel in the OT while he was in Hades (the part for the righteous not for the evil ones) after his death.
And Samuel said to Saul, Why hast thou disquieted me, to bring me up?
And Saul answered, I am sore distressed; for the Philistines make war
against me, and God is departed from me, and answereth me no more,
neither by prophets, nor by dreams: therefore I have called thee, that
thou mayest make known unto me what I shall do. (1Sam 28:15)
Samuel was disquieted while he was in Sheol or Hades (in the earth), this also implicates a conscious change from comfort to discomfort.
Hades part for the righteous ones (who are made righteous by the blood of Jesus Christ not they are righteous themselves), was inside the earth but it is separate from the part of the evil ones, and they were taken by Jesus Christ to paradise above first after His death on the cross.
Another evidence of consciousness and knowledge we see in the Psalms:
Consume [them] in wrath, consume [them], that they [may] not [be]: and
let them know that God ruleth in Jacob unto the ends of the earth.
Selah. (Ps 59:13)
Here, let them know, indicates there is no thought of annihilation or ceasing to exist.