yes, easily. the problem with the original is that there are three rooms with an odd number of doors
if you pass through a room's door an odd number of times then you will end up inside it if you started outside, and outside it if you started inside
to do that with three or more rooms is impossible, because you will always start outside at least two of them, and you can't end up inside both of them
if you're going through every door twice, there are essentially two doors in every wall with a door in it, which means there are no rooms with odd numbers of doors. you can start in one room (and outside all the others) and end up in the same room (and outside all the others)