Since the bearing surface inside a collet is relatively short, I wouldn't recommend having anything other than solid shank held by a collet. Except for end mills that happen to be the o.d. of the shank (3/16", 3/8", or whatever), the flutes wouldn't contact the collet (e.g. 5/16" end mill with 3/8" shank), or they wouldn't fit (e.g. 7/16" end mill with 3/8" shank).
If your setup isn't rigid enough, think about getting a stub end mill...or taking lighter cuts.
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