KMC 9-speed links are listed as 6.6mm pin length. KMC 10-speed links are 5.90, so if all else fails I guess you could try one of these instead, as long as it closes properly. But maybe you just have to tweak the limit screw slightly to 'centre' the chain (or at least centre the quick link pins!) on the sprocket? You've done nothing else to change anything, provided you've re-assembled the cassette correctly, and you're sure you have.

Sounds like you just have to compensate somehow for the wider pins, if possible. Try adjusting the limit screw anyway. If it's the smallest sprocket / highest gear we're talking about, you may be as well to slacken the cable right off, then re-set the cable tension and the indexing after you've adjusted the limit screw. Failing that, try the 10-speed quick links - I'm fairly sure 10-speed chains run ok on 9-speed cassettes, but maybe not vice versa, although someone who knows more than me could confirm that (none of my bikes can count past 8 - they're like, 6,7,8... um... lots!).

You can check the specs on these links (no pun intended!):
Halfords | KMC 9X Chain-Links Card of 2
Halfords | KMC 10X Shim/Sram/KMC Bike Chain Links - Card Of 2