Old Veneer

I recently came into an old Singer treadle sewing machine. Most of the veneer is in fair to good shape, but is delaminating to various degrees from almost all the way off to just the edges being loose.
I am seriously contemplating refinishing it (as opposed to using the base for a hall table).
Anyone want to give me some suggestions on re-gluing, especially those barely loose pieces, with only a 1/16" gap or less to get glue into?
TIA Bill

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