It seems that you are having a problem with the needle clamp screw on your Singer 319K sewing machine. Unfortunately, none of the sources I found directly address this issue, but here's what I found that may help:
+ The 319K was manufactured between 1956 and 1961 and it uses Type 206x13 needles, which have a shorter point than the classic 15x1 needles used on most other models. It is important to use the right needle for the machine to avoid any problems.
+ The needle clamp is usually held in place by two screws - one is an Allen wrench and the other may be a flat head screw. It suggests that if the needle keeps falling out of the machine, you can try lining up the needle clamp with the J-shaped part that is inserted into the clamp in a small hole on the bottom of the clamp.
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For full context, visit
https://www.inspirepoint.com/sewing/needle-clampscrew-657941-.htm