The first direct link I found to Abercrombie & Fitch was in 1949, when the Solunar was launched at their instigation:

http://www.calibre11.com/history-heuer-1930s-1940s/4/

That's quite a big step at the behest of a stockist, though, so I imagine the relationship between Heuer and Abercrombie was already pretty strong at the time, probably having been built up over a few years.

The watch looks pretty much spot on for a 5408 ATT, just with A&F rather than Heuer branding:

The earlier models would have had bar pushers like your watch. Nice find!