This might be a DirectInput issue, as Jazz2 was a game made many years before xinput was a thing. But if I don't power on one of my wireless Xbox One gamepads before launching jazz2.exe (with Jazz2+ installed of course), the inputs from my Xbox controller won't get registered at all in Jazz2+. I would then have to exit the game, and power the Xbox controller on again before launching the game if I would like to play this game with a gamepad.

Maybe there's limitations in DirectInput itself that doesn't allow for a gamepad to be initialized after launching games using DirectInput as the controller API?

(I did afterwards do another test, just to check if an initialized Xbox gamepad in Jazz2+ would permanently loose connection in-game after pulling out a battery inside, and putting the battery in again a few seconds later and powering the gamepad on once more to see if the Xbox controller got re-initialized in-game. To my surprise, once a Xbox controller is initialized before launch, if the controller looses connection because of empty batteries, the game still remembers the gamepad after changing batteries.)