Your watch looks original and complete to me. The hands look correct, and HF did indeed make steel cases for Tissot. The crown probably has been replaced. Your watch dates to about 1955, which along with its full rotor automatic movement, places it firmly in the "second" generation of watches with the Militar model name, produced between 1954 and 1958. I think you will find it to be a lovely watch to own and wear regularly. Here is mine from that same second generation:
There also was a "first" generation of Militar watches, produced between 1943 and 1945. Many of these were indeed consigned by the US military. Mil-watches will be engraved on the back with a serial number. All of them, military or not, were manual wind watches with the Militar model name at the top of the dial. Here's mine, with its back stamp, from 1945:
Some vintage advertising of the first generation Militar:
In the early 1940s, before the Militar name was ever added to the dial, some watches were advertised as "Militar." These watches all came with a factory signed shrapnel guard that makes them very cool to wear 😎 but awfully hard to read:
