that puts me back with the .22 Hornet, although I had considered others like .204 & .222 and still would not rule any out just yet.
The Hornet is a lovely round, but limited. Delicate to reload, the cases are thin, fragile and results can be variable.
222 or 223 are solid reliable calibres and perfect for reloading. 222 probably the best, with longer neck, but typical 222 barrel twists (14") will likely limit you to lighter weight bullets. 223 (12" or faster) gives more options.
For the future, Hornet is not deer-legal. 222 and 223 are, for small deer.
You can download a 222 or 223 to Hornet level with great results (use the same charge), but you can't do the reverse.
So I suggest 222 or 223.
.20 Vartarg, extremely efficient round (read cheap), only 18gn of powder will push a 32gn bullet over 3800Fps, the 40 grainers will be good for foxes and are still travelling at 3500fps. You could do a light load for the rabbits if you wanted. Doesn't heat or burn your barrel like a big .22 will and the ballistics are way better than the .22's unless you go right up to 70 grain bullets which will need a big case and lots of powder. The .20's are much better in the wind.
I have done a calculation that showed I was sub 40p a shot with mine...