HH is 38cm tall Alaska is about 46cm tall
. The listing for the HH says 50cm tall but that includes the height of the plastic penthouse (which really needs removing!). The actual cage itself is 38cm tall (about 36cm tall inside).
The HH works best just as a plain cage with one shelf left in and the Penthouse and tubes removed (it comes with caps to block off the tubes). It is a good size and has a huge door that goes right to the top of the cage - one of the best things. Plus low height. You can sometimes find them second hand on ebay cheaper although with the current situation that might not be easy or a good idea - not sure.
Also forgot to add - at four months old your hammie is still a child - very young - easily scared. I'm sure he'll soon settle down.