My reply: I base his omnipresence on these things:
1. Theologians much smarter than me have said he is.
2. References in scripture. One search I just did said that there are100s of passages which speak of the ways he is all and in all.
One author suggested his top ten: My favorites among them were Jeremiah 23:24; Psalm 139:7-10; Proverbs15:3; Isaiah 57:15.
3. However I also based my conclusion on a personal study of Genesis 1 and some other clues. In Gen 1 I first realized God is an invisible Spirit but he is able to establish places and things within himself. These would ultimately be manifested and put to use.
One might compare this creative process to how an architect would mentally think of a plan for a shopping mall and then draw out the plans and provide the supplies and build it ... EXCEPT for God this is the truth: God's spiritual essence is life, power, intellect and so forth; so, God can create things within himself and give measures of life to what ever living beings he wishes to manifest. Thus, I believe male and female mentioned in Genesis 1 were living within the invisible spiritual essence God as two unique spiritual beings in the beginning ... but they were body-less. Architects can't give life to what they create!
Then in Gen. 1:6 ... I noticed God was creating realms or places where things, like the sun, moon, stars etc, would exist. Next I noticed the dwelling place of earth... and upon it there were specific places where things could dwell, the birds of the air, the sea and all that was in it... etc. Much later in the NT I learned about other places established within God called - Abraham's bosom, the Pit, Hell, a place in the light versus the place in the darkness for lost souls, and the Lake of Fire. The essence of God is in all those places for he created them and they dwell within him. And remember he even knows the number of hairs on our heads and I've figured he knows how many are on the ground. And let's not forget he sees us forming in the womb and he sees into our hearts and know our spiritual intentions. How much more everywhere can he be?
Isaiah 43:11 tells us the promised one will only be: God, LORD, and Savior. I think Col 1:16-17 is a great verse for adding a nuance to what God's relationship is to creation. It is talking about Jesus, but we know Jesus was God, LORD, and Savior finally manifested upon the earth in flesh doing what God told him to do and say. Verse 17 explains he, being God/Emmanuel, was before all things and by him all things consist.
Now if by him all things consist - notice it is talking about what things consist of NOT about what caused their existence... therefore, God's essence must be some part of the essence of all things that exist whether they be thrones powers, principalities, etc.
Now, what does what I have written have to do with all things being in the Spirit of God? You know Jesus our risen Lord is something like a dwelling place... for we believers will dwell in him safely because he is in us. We will be one with our Lord. What do I mean? In Genesis 1 God said let there be LIGHT. John 1 tells us THAT LIGHT had come into the world and he was called Jesus. Other scripture reveals we believers shall well in the LIGHT versus being cast into outer darkness. These are places/realms within God. Believers are no longer programmed to dwell in outer darkness but with our Lord in the LIGHT. See how God separates things and places things ... but it all exists within Him.
Well, I rambled in this part of my answer, but there are just so many bits of clues that for me do support HIS omnipresence.