Small part work holding?

A little background for my question, I’m working on a HAAS ST20y, which the smallest part I can get jaws for on the sub spindle 3 jaw will only reach 1/2” diameter. Made a set of jaws which could grab the part I’m turning to .250”. But I’ve only got 3/16 to hold onto, which only spits the part out once the tool makes contact. So my question is. Are there any options which would not require me to get another collet chuck for the sub spindle. It’s currently equipped with a 5” sub spindle.