Stainless steel (depending upon the grade) isn't necessarily harder, but it is tougher. LOW RPM and high pressure, and cutting oil of some kind. I like TIN coated 135 degree split point bits for tough materials. Stainless will also work harden, especially if you go at it with too high an RPM. What grade of SS are you drilling, and what is the drill bit diameter?
Ayuh,.... Agreed,.... for 5/16", I wouldn't pilot-hole it,....
Probably try a carbide bit, 'n lotsa oil,....usually Atf,...
The work-hardenin' can ruin a hole,...
If yer bit ain't cuttin',... Quit,....
Get another bit, resharpen, whatever, 'n pray it'll pop through the hard area,...
Lotsa oil, slow speeds, 'n steady chips,...