There are no secret or hidden levels in PS1/PSX Final Doom;
and subsequently there are no "secret" exits, per se.

Level 23, Ballistyx, has 2 exits, one that's accessible in
Single-player and Cooperative games, and another exit (just
outside of a caged area which overlooks a walled-in field
of pedestals) that's accessible only in Deathmatch games...
What you're seeing is the Deathmatch exit.  (In Deathmatch
games only this outside part of the map is accessible.)

Both exits lead to level 24, Heck.

For more information including maps and screenshots, access
http://classicdoom.com/bal-dmx.htm