No don't repaint....if you want the cookie jar to look new, buy a repro.
But I don't know of a plain green glass that had a metal lid....could it be that there was a glass lid that got broken and was replaced with a tin lid from something else? In which case since the lid isn't original it's wouldn't hurt any value to repaint.
Can you post a picture?
Linda C

Oh yeah....I've seen them and the lids were meant to be metal, because they screwed onto the glass jar. I don't know who made them, but they always struck me as looking like something Hazel Atlas would make. Do not.......nooooooo.....do not paint that lid.

Kitchen Glassware of the Depression Years by Gene Florence 3rd edition has large sugar container that is called Zipper Pattern, It is round looks almost like a Halloween pumpkin with the 'zipper" ribs going up the sides & metal lid. It would make a perfect "cookie jar" Manufacturer was Sneath Mfg. Co."Quaker" pattern (called zipper by collectors) it was priced at $85 in 1987. There was a pickle jar that was taller with metal lid & bail type handle for about same price. It was more "pickle barrel shape" only 2 similar to what you sound like you have.

Thanks, I decided not to repaint it. The top is original and it's probably made by hazel atlas.