Most "off the rack" bonsai pots are less than 4 inches deep interior dimensions. My recommendation would be the next pot be a training pot shaped more like a "finished bonsai pot". Myself I use Anderson flats, a heavy duty black plastic flat with a mesh bottom. Their interior dimensions are 16 x 16 x 5 inches, and I usually fill to only 4 inches depth with potting media. This size container holds nearly the same amount of potting media as a 3 gallon nursery trade pot. (note, the "3 gallon nursery pot" holds less than 3 gallons by volume). The Anderson Flat is my "go to" for a growing out & training container. There are hundreds of alternatives, it is up to you to choose one, but it is a good example of what I use.
In the images you can see my Anderson flats in use, first 2 images for seedlings (persimmon) and the third image for 2 trees in training, a spruce and a pine. Like I said, brand name, or specific style is not important, but the next training pot should have good drainage, be wide and long, but less than 5 inches deep, and generally you should not have potting media greater than 4 inches deep, unless you can not reduce the root system adequately to make the tree fit.
View attachment 321616 View attachment 321617 View attachment 321618