Unfortunately, except using your vehicles, there is no other way of quickly moving around the districts in Mafia 3. So, when you get tired of looking at the same backgrounds, then you should install the TBD mod for fast teleportation between districts.
It is really easy to use -- after you install the script just press the corresponding keys on your keyboard to teleport Lincoln Clay to any of the 10 districts in the game. Here are the key bindings:
Note: These won’t work on your numpad, as the creator of the mod didn’t want his mod to interfere with the rest of the mods in this selection that mostly use the numpad keys.