I work at a very large company that is well known in the industry.
The problem is that sometimes we don't have enough conference rooms in the facility
These two statements don't play well together. It sounds like there's a bigger problem to solve here: either your company does not have sufficient facilities for its needs, or your co-workers are misusing conference rooms, for example as working spaces.
For your current situation, the best thing is if you can get some general consensus that interviews are a high-priority use of space, and should bump other meetings. If this is not possible or if you have trouble getting it implemented consistently, I would suggest you ask to be allocated funds to hire a suitable room, for example at a co-working facility.
There is a useful lesson in managing up here. It is quite plausible that if presented with the three options, "have a meeting in a hangout space", "give interviews priority for conference rooms", or "hold interviews offsite" they will choose option 2 and take steps. If option 3 is not part of the package, they will choose option 2 and not take the necessary steps. This is why I recommend including option 3 - not because I think it's a good option, but because including it will force action.
