Same here (and I have been saying so for years).
Our reasoning is, of course, pure Calvinist-Protestant Sola Scriptura -- the Scriptures, if taken by themselves, speak of the Spirit as proceeding from the Father and being sent by the Son (John 15:26).
Thus, if one is to speak of the Holy Spirit "proceeding from the Father and the Son", the only explicit affirmation which we are able to directly derive from Scripture (as regards this admittedly high and mysterious Trinitarian economy) would be the affirmation that the Spirit issues from the Son in a mediatorial procession from the Father, as affirmed by Saint Gregory of Nyssa.
So, I guess I'm not the only FR Calvinist who (based upon Sola Scriptura) "leans towards the Greek" in my understanding of the Filioque.
But, yeah, that's a subject for another Thread.
By this logic, OP, one could then say that the spirit "proceeds" from others in the "mediatorial" manner when you consider Matthew 10:20 "for it will not be you speaking, but the Spirit of your Father speaking through you."
Orthodoxy tends to stay with the basics. The Divine Economy is eternal, thus the Word is eternally begotten by the Wisdom, and the Spirit eternally proceeds (wells) from the Wisdom. That never changes, always keeping in mind that God is transcendental.
Instead, you are thinking of the Pentecost -- a singluar point in time. Open up a little more; you are getting there. :-)