In my wanderings through community bands and pickup groups, I often see players suffering with, or almost unable to play, certain fingerings and fingering progressions because they try to use the fingerings they learned from standard fingering charts.
Here are five fingering alternatives that will likely not only be easier, but that may make some "impossible" fingerings (especially in rapid passages and tremolos) manageable.
C# above the staff - First finger of the left hand only (no thumb or register key). Particularly useful when playing a tremolo between this note and the notes below it. I've used it mostly for tremolos when I have to use a a RH key for the lower note.
D above the staff - No fingers, no thumb, no register key, just an overblown "open G". Somewhat flat, but not enough to matter if you're going fairly fast.
Bb above the staff or Eb on the bottom line of the staff- First and third LH fingers. Very helpful in tremolos between G above the top line of the staff; all you need to do is move one finger! (This is a standard fingering on German system clarinets, but it will work on the Boehm system in a pinch, though the tuning suffers.)
For the usual "1 and 1" fingering for Bb in the clarion and Eb in the chalumeau, you can use any of the three RH fingers for these notes; you're not restricted to just the first finger.
For F above the staff, six fingers and the LH G#/C# key (no RH G#/D# key). This is called "the long F". It speaks a lot better than the standard F fingering, is usually better in tune and has a nice quality, and doesn't "jump out" like the standard F does. (On some clarinets, adding the RH G#/D# key will give reasonable F#, which comes in handy if your instrument will render it.)
There are scads more "fake fingerings" on the clarinet, but these five are the ones I use the most and that I suffer when I see other players not taking advantage of them.