Denny, the KeyMan checklist is a good start, but it is far from perfect. For instance, the set listed for 1953 is missing Ed Lopat and, surprisingly, Mickey Mantle. The 1953 set is an "Action Photos" set, consists of 12 photos, and is 8"-by-10". There are also "Action Photos" sets for 1955, 1956, 1957, and 1960. I suspect there are sets for 1954, 1958, and 1959, although I haven't seen them. I've attached a photo of the envelope for 1955. The Yankees also issued team sets that are 6"-by-8 1/2". I have sets that are from 1956, 1958, 1959, 1960, and 1961 or 1962. I suspect there is probably one for 1957 as well. There are 23 photos in the 1956 set, 16 photos in the 1958 and 1959 sets, and 14 photos in the 1960 and 1961/2 sets. I've attached photos for the 1956, 1959, and 1961/2 envelopes.
Another place I prefer for photo set checklists is the following:
https://www.tcdb.com/ViewAll.cfm/sp/Baseball/year/1961
You can find checklists for the '61 photo sets at the link and can rummage around for the other years as well. The TCDB lists the sets that you're referring to as "Team Issue" as being team issued by "Sports Service." A nice thing is that photos are included for some of the sets.
One issue with the Jay Publishing and Sports Service sets is that they may include different players in the 12 card set during a given year. The TCDB will list these as set A, set B, and sometimes set C for a given year.
In 1966, the Yankees had a color set of 8-by-10 photos that they gave away as a promotion at a game and also probably sold at the stadium. In addition, they had a similar set in 1965. The only difference in the sets is the 1965 set had Johnny Keane and the 1966 set had Ralph Houk. It seems not too many people are aware of the 1965 set, but TCDB lists them both under the appropriate years.
Even though I prefer the TCDB checklists, they are not perfect either. For instance, they do not list the "Action Photos" sets or the 6"-by-8 1/2" team issued picture packs.
Lastly, I don't think we'll ever see a definitive checklist for Yankee photo sets, but I think the TCDB and KeyMan lists make a great starting point.