You'll find out soon enough when you go this route though at times painful is best in the long run. Your mtrbd. should be the mini-ATX type and most better PC cases provides multiple mounting point for various mtrbds. made. They follow the typical sizing, however since this is a Dell(OEM type) that's not a given. However, what you're doing has been done before at least for other models. In most cases, google for an answer because common OEM systems are swapped to generic PC cases often enough, maybe yours too. If your PSU is the "slim type" yes you will need to replace with the generic type PSU for your new PC case. Practically, everything else can swap over. You may want to verify the new PC case rear bulkhead that provides the "holes" for posts and such matches what your Dell 660 has, if not just leave the "removable bulkhead plate" off.
The real benefit besides if it works with some effort, it that when you need to upgrade/update or renew again, your PC case will handle it hard easier when you 1st got it. replace the mtrbd. at a later date for a better one or keeping up, etc., will be the bonus of all this. The PSU is generic size, so when that gets too old, the replacement is a "drop-in". The other main benefit of a decent PC case is cooling, more mounting areas for new fans and the larger case allows better "airflow circulation" for any added heat as time goes by.
Currently I've got an Inspiron 660s.
It's a slim case, so i'm at the point where I can't really make any worthwhile upgrades to the system due to space constraints. I want to buy a new (bigger) computer case, but I'm not sure if I'll have to replace any other parts.
Will I need to buy a new motherboard?
I would imagine that I need a new power supply, which is fine, but I just don't want to buy a new case to find that I'll have to basically rebuild everything but the HDD.