I bought one of those pillows too, unfortunately it was for our bedroom vent that has a fantastic fan in it, and it barely fits, and falls out over time. I bought it to block out the morning sun, so keeping the morning sun out, but gave up on it, and finally just painted the black vent cover white which works good. I would not consider keeping the pillow in the vent during storage. I believe it would do more harm than good. I open 2 vents both with covers during storage
Proud father of a US Marine