What can I use for interior helmet padding? My bike helmet needs more padding on the inside.

Any suggestions of what to use?

I"m thinking double sided tape and some sort of neoprene or other cushiony material.
I have used some half-inch thick strips of foam to help raise the helmet

You need to add the foam where there is a void between your scalp and the helmet.   It should not be added where the helmet presses on your head.

Small amount should not mess up how the helmet works.  Its no different to going from a long haircut to a short haircut.

In winter I wear a thin beanie under my helmet.  Some people wear bandannas or cycling caps.

If its a really bad fit you might want to replace it.   Why did you buy it if its a bad fit?   Do not mail-order helmets site-unseen!!!