This tiny pin, a little smaller than a nickel, is the only example of "TAD" merchandise of which I am aware. It was a premium for Tokio Cigarettes, and I've seen references to these being from about 1912, but the military uniform makes me think it's from WWI, probably circa 1914.
What the heck does it mean? I have no idea. Is it a foreign policy gag? Are those military encampment tents in the background, or are they teepees? If the latter, is it a joke about the cavalry's relationship with native Americans? Such a political statement would be uncharacteristic of Dorgan. More likely this is just yet another bit of Tad slang whose meaning is lost in the mists of time (UPDATED, see below).
Cartoonists and syndicates had a rather laissez-faire attitude towards merchandising in the early days of the comic strip. The syndicates' attitude was that no publicity was bad publicity, so they took little notice of companies making use of their properties, figuring it would draw traffic to the newspapers, and they certainly took no interest in controlling such use. Some cartoonists enthusiastically embraced the commercial opportunities merchandising presented, particularly R. F. Outcault, who made a fortune from Buster Brown's popularity. It wasn't until the Popeye and Mickey Mouse juggernauts hit in the 1930s (after Tad's time) that the syndicates took notice and asserted control over the gold mine they previously didn't even know was there.
I'm indebted to Richard Marschall, whose essay on Popeye merchandising in the last volume of Fantagraphics' essential Thimble Theatre reprints for the information above.
UPDATE: According to Pony Pal Mark Newgarden, Tokio issued several Tad pinbacks. The phrase was used on many different designs, probably a popular slang term at the time, and may not have had any direct relation to the illustration at all. Thanks again, Mark!