Um, take a look at a few posts above (reply #125), geoo described exactly the same thing you're seeing.
I can tell you that if you make the pixel at (580,110)
not terrain [the bottom-left corner pixel of the steel block right above the one you are climbing), the problem should go away.
Probably not worth explaining how it comes about, but basically, there are basically two things the game needs to check a climber for:
1) whether it has reached the top of the wall, and therefore can start transitioning back to walking
2) whether it hits its head on an overhang and should fall
You'd think that the game would do both things on each frame where movement occurs, but instead, it's more like during 4 animation frames it does #1 only, and then during the next 4 frames it does #2 only, and so forth. And also it only changes the lemming's vertical y-position during
either #1 or #2 only
(forgot which one off top of my head), but
before doing the "hits head" check. The basic end result of this strange, incomprehensible order of processing is that, depending on height of climb, in some cases a climb would succeed despite an overhang, while at other times a "false overhang" like the one you have could still abort the climb.
What can I say, climber mechanics in Lemmings is full of strange behavior. I believe the very first post of this thread already listed like 2 or 3 more well-known glitches about climbers......