Sometimes doing just one change won't help so in this case you would do another water change around 50% and test again. Your tank is now trying to cycle itself with the Betta in so you'll see ammonia jumping around like that as well as the other levels of nitrites and nitrates sometimes.
I don't remember if we had gone through the cycle process with you at all? There's a really good sticky in the Betta Habitat, tank and bowls section I believe or it's in the Betta Care.
But anyway, your Betta will be fine, a little stressed, but he'll be fine as long as it doesn't go up to like 5ppm or something! But for now just keep testing every day or so to monitor the ammonia and when it gets above .25ppm, do a water change or a few of them to help out. And yes, you can totally add more plants, the more you have the better and easier they will keep the ammonia, nitrites and nitrates at bay