I've even tried a few "what if" conditions (assuming another cell is filled in a particular way, and seeing what can be deduced from there), but I haven't managed to deduce enough to fill in any more cells for sure. What am I missing?
Look at the second column and try to place the 6.
Row 1/4 - cannot be 6 since row already has a 6.
Row 1/3/5/6 - cannot be 6 since number must be less than another digit.
Therefore, 6 must go in row 2.
This leads to some immediate deductions in the right-most column, and the whole solution follows by the usual rules of the puzzle.
If you still want to solve the puzzle yourself, do not view the next spoiler!