I think you guys are overcomplicating things a bit. I have little doubt that they were likely referring to T3's.
The Turkish Cigarette brands were ATC brands and they likely didn't mind cross-advertising other products. Helmar and Turkish Trophies were very similar brands and in some cases even produced the same series of cards at the same time (T52 Costumes of All Nations). Also given the fact this is an "outside of the package" ad sticker I don't think it is outside the realm of possibilities that it was simply a test promotion or a hastily put together one (to the tune of giving out a premium with a Turkish Trophy box that doesn't specifically refer to that exact brand), also keep in mind that what we call "T3 Turkey Red" cards were also advertised and given away just as much with FEZ and OLD MILL cigarettes as they were Turkey Red, we only call them "Turkey Red" cards because some of the second series had that advertised on back without the checklist, but had the second series of cards not been produced they wouldn't be called simply "Tukey Reds" as they advertised the other brands just as much as the Turkey Red brand. Another thing to keep in mind is that maybe the ones given away with this promotion are the non-advertising "checklist only" cards that are rarely seen today.
A similar type of promotion was done with the Hamilton King (or T7) girl cards. We think of these as "send-away" premiums but I found a VERY brittle envelope some years back containing a complete set of the "Flag Girls" that had instructions for the retailer very similar to what was probably furnished with the shipment of "ball player photo prints" that we see on the Turkish Trophies boxes above...
