I don't know if there is a way you can increase your storage space, however you could try using Google Voice. As far as I know, there is not a limit on the amount of storage space you have. At least I cannot find anywhere which states a limit.
No way to increase voice mail space? I have a device with an ungodly amount of space, and I can't increase it? Another reason why companies lose business. I just upgraded to an iPhone 7 Plus thinking it's better than my iPhone 6. The bluetooth can't hold a signal, and calls drop for no reason. Exchanged it, and the new phone does the same thing. They should be recalled since there are thousands of people that are having the same issues according to the number of users on this forum. Hoping that's an "iPhone" thing and not a "Verizon" thing.
Can't use Google Voice either. Not permitted due to corporate policy.