While I don't have hydro assist, I do know my PS gear *loved* the addition of a cooler on the return line!
The fluid used to get hot after an hour (sometimes quicker) on the trail ... pump would groan & whine, steering effort would fluctuate between normal & "Popeye" in a few seconds and the lines would be nuclear hot.
I threw a cooler (8"x12" Ferd tranny cooler from JY) on the inner fender, plumbed it... Steering stays consistent, noise from the pump went way down, & the lines are noticeably cooler.
Could an aux (*sealed*) reservoir be plumbed in on the return side? Pump> box> aux reservoir> cooler> Pump without any issues like air bubbles, etc? Seems the return pressure would be enough to keep it all bled & flowing? (Zero room to extend the the current ham-can)