Labyrinth cool game mechanic, but weak game

Background

Labyrinth, the board game, has a sliding tiles mechanic that I really like. Sixteen tiles are fixed on the board and 33 (7x7 - 16 + 1) loose tiles are available to slide through the frame created by the fixed tiles.

This picture shows the board and J3 punching out the loose tiles:

Here, we've started to put some of the loose tiles into the board:

You can see that each tile has a picture that includes a segment of a maze and, possibly, an item. Paths through the labyrinth depend on how the tiles are arranged. Along the open rows and columns, loose tiles can slide up/down or left/right. This is also where the extra loose tile comes in: sliding a column creates an open space for the extra tile and forces another one off the board.

My problem

We don't find the game very compelling, so I am looking for ideas about how to make use of the underlying structure in a new game. We have done some investigations about arrangements of the loose tiles (can we create a maze without any inaccessible tiles? What do we get when we minimize/maximize the number of inaccessible tiles?) However, I feel like there is a lot more that should be possible.

What are your ideas?

An unrelated project about birds' wing shapes from J2: