I'll start with inside measurements:
front to back 1 3/4" approx.
side to side 1 1/2" approx.
The hat could be a bit bigger than these measurements and you'd be ok.
Depth is a bit trickier. There are the 2 high points and a very low point. I'll go from the middle of the low point which is a little over 3/4 of an inch in back and front almost 7/8". (The hat is longer in the front than the back.)
These measurements are very general and I hope they help!
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