Holes.

[Triple J]

I'm thinking of something like a tiny version of a level splitter, or something like a door that doesn't need to be clicked on.
These things could be recolored, resized, and rotated to fit in walls and ceilings, or through the ground, and entering the hole could act like an "automatic door." You choose the destination, and the hole will take you there.

This may be asking for a lot, but could the holes POSSIBLY morph with tiles? It might make it easier to avoid cutoff.

[AwesomeJRFD]

This is what the "Move Character" trigger function is for.

[Triple J]

Oh, right. Why haven't I thought of that? :3