Does authenticated encryption add any protection to a signed digital envelope? The general scenario of hybrid encryption is to use public key cryptography for the exchange of a one time session key that can be used for the encryption of the plaintext with a symmetric encryption algorithm. A digital signature attached to that structure is what I know as a digital envelope. Furthermore, some resources agree that sign-encrypt-sign can help to prevent certain unlikely but possible attacks.

On the other hand, for strictly symmetric scenarios, the use of authenticated encryption is usually recommended as a default good practice.

Does it make sense to bring these two approaches together? Especially, would authenticated encryption (as opposed to semantically secure but unauthenticated encryption) provide any protection against any scenario not covered already by the digital signature?
I suggest that you look at Signcryption; a short survey appears here, and efficient schemes appear here. Just signing then encrypting or vice versa in a naive way is not secure (especially in the multi-user setting). So you have to do this right. Once you have a concrete scheme, you then have to see what level of security the encryption scheme needs to be. Typically it does need to be CCA secure, and the way to build CCA-secure symmetric encryption is the same as authenticated encryption. (Note that doing hybrid encryption or KEM/DEM, you need both the asymmetric and symmetric parts to be CCA secure.)  However, it really depends on exactly what you are doing. Look at the cited references as a starting point.