There are some interesting variations between this and the version I learned as a kid:
In the version I learned growing up, there is a separate stack with your winnings. When you run out of cards, you shuffle your winnings and then begin playing them.
In the version I’m familiar with, you deal three cards down and then another card face-up. As I’m writing this, I see @bh mentioned something similar.
When I’ve played, the second time you do this it is called a “double war”, the third time it is called a “triple war” etcetera (and if you played with three people, a two-person war was called a two-way war and a three person war called a three-way war).
In my case, I don’t remember exactly what would happen. I think that you just sent fewer cards into war, as you described here (unless you had none left, in which case you just lost):
Upon playing the game, an additional variation I noticed was that you don’t get to see the three cards you won or lost in a war. I understand that that might be harder to program (I haven’t looked at how you did it yet), but it is a very different dynamic. In the game I played, I started with the Ace of Diamonds. The only way you can lose an Ace is through a war (or if you play with Jokers). At a certain point along in the game, the computer flipped the Ace of Diamonds, showing that it had abducted it without my knowing. This has broader implications for the overall game experience for me.
When I used to play, even though there was no strategy I could use, I think it helped my strategization (somehow that isn’t a word) and memorization skills. Specifically, I would pay close attention to how many Aces each player had in the start round; this was very easy because you could know when you’d made it through the deck because of the aforementioned shuffling when running out of cards. Thus, once the first hand had been played, I knew exactly where the power lay (sometimes I’d even count Kings). I’d keep track of any developments for the rest of the game. I’d also pay attention to the 2s, especially in the later game. If I was losing, 2s were fluff cards that just added to my card count and would almost certainly be lost. However, if my opponent had them that wasn’t a bad thing, if I won a 2 that meant I didn’t lose any cards. Additionally, as a winning opponent, I hated having the 2s and giving the losing person more time and ammo. One of the big things I’d “strategize” in was when I was losing. In that case I knew exactly what I had and what my high cards were. So, every time I shuffled, I knew what kind of order I probably wanted (especially if I was shuffling mid-war). And, of course, I was always hoping against hope I would win an Ace in a war. As a losing player, Aces are far more powerful because you reshuffle more often. Thus the game is often determined by who has the Aces; sometimes we would even separate the Aces and give two to each person to shuffle into their deck.
Adding Jokers was always an interesting variation. They upset the dominance of the Aces by trumping them, but as there were only two Jokers the dynamic was a bit different. The Aces remained quite powerful even as the Jokers could beat them. However, I think it perhaps best that they weren’t included as an option…
Me! I’ve been wanting to make a version of Hand and Foot game which you can play on the cloud for a while now (no promises though).
