When Jesus was baptized by John, and the Holy Spirit descended on Him like a dove, was the Holy Spirit not with Him up until then?
Short answer: No. Not as an empowerment for ministry. Up until that moment Jesus had done no miracles, preached no sermons, etc.
This is consistent with the depiction in the synoptic gospels.
Longer answer: Define "with"!
The Spirit descended and stayed upon Jesus. The accounts about Jesus baptism do not give us much information about what one might call the "metaphysics" of God, i.e. how the persons in the trinity relate to each other eternally.
(The only major thinkers that thought this was the case in early Christendom were the adoptionists.)
All 4 gospels seem to see Jesus' baptism as the transitional event into public ministry. But since there also is an affirmation of Jesus' sonship, one can safely assume that it also was an expression of how the love the Father has for His Son is expressed, through the spirit.
Traditional theology - for God reasons - say that this continual expression of love within God, did not begin with Jesus' baptism, but is in fact eternal.
The empowerment for "Kingdom of God service" in the power of the Spirit, given to the incarnated Christ, began at this moment, however.
I'll refer you to Summa Theologica 3.39.6,
Furthermore, consider that God (Father, Son, and Holy Spirit) are fully present in every place once can be said to be present. Thus, Christ is never without the presence of the Father and the Spirit. It is only for the purpose of sanctifying the human institution of baptism (John's baptism) that Jesus "receives" the Holy Spirit which is already with Him.
And as addressed in Question 6 of the same chapter of Summa Theologica, Christ is not so much receiving the Spirit for Himself, since the Spirit is already fully with Him; He is receiving the Holy Spirit into the sacrament of baptism.