Yep, according to ISMACS it's a 66, 1 of 50,000 allotted August 8, 1922.
I have purchased several reproduction slide plates for 66 type machines and they've all fit. The finish isn't a good match, and on one the edges had to be dressed down. But unless you want to pay eBay prices for an original they are a good buy.
Before buying a new belt or adjusting the one there I'd oil the machine thoroughly and the treadle irons too. That will make a big difference.
When I adjust a leather belt I take the staple out, drill a second hole a 1/4" back from the one there then trim the belt at the old hole and put the staple back in. It doesn't take much to tighten them up. They don't have to be real tight, just tight enough it doesn't slip.
Congrats on the new to you machine. My first treadle was / is a Singer 66-1 red eye. It works great when I can get it unburried to use it.
Joe