Rigidly rhymes with:
admittedly, assertedly, candidly, discontentedly, evenhandedly, good-heartedly, guardedly, handedly, heartedly, heatedly, mindedly, placidly, pointedly, purportedly, repeatedly, reputedly, singlehandedly, unabashedly, undoubtedly, unexpectedly, unprecedentedly, validly, wholeheartedly, wickedly, wretchedly
Rigidly sounds like:
ragsdale, recital, rectal, reichstuhl, reschedule, resdel, resettle, residual, restyle, rhizoidal, riccitelli, rightly, rookstool, ruggedly, rustle