What software do you have and what type of machine are you using?  At 3mm the letters would sink in on a sweater.  I don't think you will ever be able to use a true type font that will sew quality lettering at a size less than 6mm.  I have created some small fonts for Wilcom ES keyboard lettering that I sell (ESA files) that sew great at the smaller lettering size of 5mm and 5.5mm with standard 40 weight thread and if you use 60 weight thread it sew even smaller.  Other wise you have to digitize each letters to get quality sewing.  I just got done doing some digitizing of letters with 60w thread on a knits pique at 0.1 inch (that’s ~2.5mm I think) and it looked cleaner than another embroidery lettering at ~0.2inches/~5mm on a twill.

This logo is scanned in of the sew out on a pique 4 inches wide by 1.39 inches design of a test run I had. I have edited it since then to clean it up but don't have a scanned copy of the new edited file.