It probably would help, but it would probably radically change the sound, which if you like it at the moment would be a problem.but do you think that boarding up the back might help protect the cone from excess movement?
In my school days I did a little science project which consisted of making a very deep speaker enclosure with a sliding back so I could quickly and easily change the volume. The effects were striking.
If you're not careful you'll find yourself heading down a slippery slope which involved building twenty different loudspeaker enclosures to find the best compromise for your own taste! And don't go thinking that a more expensive bass amp than the Fender is inevitably going to produce a sound you like better - I fear you just need to try them.