To find the secret exit on PSX Doom level 58, The Mansion,
which leads to level 59, Club Doom:

As with Suburbs, the secret exit in The Mansion is blocked
off, and the path is only opened temporarily by throwing a
distant switch, and then quickly running and teleporting.

Near the center of the level is a 4-way intersection: three
ground-floor paths come together in a red-carpeted T shape,
and a fourth path to the east is raised above ground level.
The southern-most path is very short and ends in a blank
wall.  Step up to this wall and hit Circle on the control
pad, and the wall lifts to reveal a small, very dark room
with 3 vertical flashing red panels.  The pathway to the
exit to level 59 is in the southeast corner of this room
(far left corner as you walk in), but is blocked off.

First go to the southwestern-most room on the map, the room
with the cage in the center of it.  As you enter this room
you'll see a red door off to the right.  You have to open
that door and Activate the switch behind it if you haven't
done so already during the game.  (You only have to do this
once during the game.  It lowers a barrier that complicates
access to a vital teleporter pad.)

From this room, walk east through the door into the next
room.  This room has a pillar in the center of a square
lava pit, and 5 teleport pads: four pads on continuously
rising/falling columns which surround the central pillar,
and a fifth pad down in the lava pit, in the southeast
corner.  On the Automap, the arrangement is like this:

#1#
2P3
#4L
    5

where 1 through 5 are the five teleport pads (1 through 4
are rising and falling), P is the central Pillar, and L is
a Lift with a switch embedded inside.
If you are down in the lava, you can Activate (hit Circle)
on L and the lift will drop down to you, simultaneously
revealing the embedded switch.  (Don't worry about the
switch just yet.  Also, there may be other ways to get to L
besides from down in the lava... explore a little. ;)

What you want to do is:

#1#
2P3
#4L
    5

- Stand on top of L, facing P, with your back to 5.  (On
  the Automap, you may see a tiny diagonal line on L.  This
  is the location of the switch down inside L.)
- Wait until 3 begins dropping down.
- Activate the switch inside of L while you stand on top of
  it.  (You can activate it this way if you are facing the
  little diagonal line that appears in L on the Automap.)
- While in Run mode (hold Square on the control pad), back
  up onto 5.
- You will materialize facing 3; Run and jump onto it.
- You will materialize in the eastern corridor of that
  four-way intersection mentioned earlier, with your back
  to the intersection.
- Step back a little to fall down into the intersection.
  The small dark room with the path to the exit is now off
  to your right.
- Run (or use Run and Strafe) to get into that dark room.
  Remember that the exit path is in the southeast corner
  (on the Automap) of this room.

Odds are, you'll hit a dead end, since the exit path closed
already. :)
The timing on all of this is VERY tight.  You may have to
practice this series of maneuvers several times before you
can do everything in one try, and fast enough to get into
the exit room in time.

Once you get into the secret exit room, you'll see a switch
and a teleporter.  The teleporter brings you back into this
level, and the switch on the wall will send you to map 59,
Club Doom.