The sticker on the TT pack states: "photo print of a famous base ball players....." and the question is what photo print is being referred to. I don't believe there is over complications here; just a question looking for a real answer. By personally having little doubt the T3 is the photo print in question is not fact. Surely there must be a better reason than guessing. It hasn't even been established whether this TT pack was produced in the 1910-1911 time frame.
Your post strays into extraneous matters without demonstrating any logical connection to your T3 opinion. T3s are premiums of Turkey Red, Fez and Old Mill not Helmar. Helmar and Turkish Trophies were not similar brands - TT was a brand of Helmar.
BTW as I recall the production of T3s were exhausted in 1911 and the message to send in coupons was removed from these cabinets in order to ease demand. There was actually a shortage of these cabinets; not a surplus to supply endless tobacco stores - even if this theory was true.
Henry - you knew that I knew. Anyway this is the cigarette pack article I sent to SCD in 2005. They made a black and white collage out of all of the images.
http://js-gallery.net/packs.htm