glue...

Deodorant screws up but it isn't round

That is actually a good idea!
Plus, it doesn't have to be round. The glue itself doesn't spin, just the thing under it

if its square than you would need a square turny thing at the bottom, that simple

These actually exist though we have some in my history class

In Australia, there a triangle glue sticks that screw up and down just like the circular ones. They are designed specifically to reach the corners. :)

Maybe you can slide it out?