Do the easy thing first, clean it well and get some high quality grease on the bolt and the especially the bolt spring/charging handle assembly. Also inspect for burrs or anyplace the bolt is rubbing on the charging stroke. Use a high quality grease like Twb-25 or Geissele purple grease. Dry lube for these areas is not enough, I know I've tried using just One Shot or Ezox in the past and it just doesn't work on the problematic 10/22's.
Your primary ejector for a 10/22 is the notch in the magazine lip, the ejector arm coming out of the trigger pack is a secondary ejector if the first misses.
2 hardware items for chasing failure to eject after cleaning/lubing:
1) KIDD 10% lighter bolt spring, by itself or in the charging handle kit $10. If you really want to cheap out you cut start cutting coils off the spring to lighten tension, but that's redneck and causes other problems if it spreads or cones.
2) stiffer magazine plunger release spring. $3 Not a magazine spring, but the plunger spring that holds the mag in the magwell. IF this is weak it tips the magazines at an angle but this is usually a failure to feed fix.
One of my KIDDS was temperamental on ejecting SV rounds for the last couple of years. Worked fine with HV, but HV ammo consistency isnt what I wanted to use in it. Spent more time clearing malfs and troubleshooting it than shooting. 10% lighter bolt spring helped but did not cure problem. Proper grease on bolt and charging rod cured problem.
Cure the semi-auto problem forever by getting a quality bolt action, that is the route I've pretty much taken.