I think there have been a few good responses already. My 2c worth...
Not sure what you class as huge (10 KLOC, 1000 KLOC, 10000 KLOC, etc), but one would hope that this is broken down in some way and is not a monolithic single program. Perhaps your management has some guidance on which 'module(s)' you are most likely to be spending time in at the moment. Hopefully this can help break down the problem scope.
Firstly, before you try to understand the code try to understand the product. What does it do? Then how does it do it? What does it interact with? Then how does it interact? etc...
When getting to the code try to understand the high level design and philosophy first, and work on the breadth before the depth. I agree with some of the above re fixing some bugs, but I also strongly suggest you continue to get a handle on the high level even if you need to get into the details to fix some bugs.
I also agree with the above in terms of generating some diagrams for yourself if you can't find any already in existence. And then share them, perhaps a team/product wiki? I'm curious as to why the existing doco does not help very much. Typically this is because this type of doco was generated from the early concepts and the product no longer bears any similarity, but if this is not the case then what can you contribute to this issue. One assumes that where you are today someone else will be in short enough order, and you are in an ideal position to know what essential doco is missing!
If the product is actually 'huge' then you have to accept that you will never be able to hold all of it in your head, so the best thing you can do is be familiar enough to know where to start looking (comes back to understanding the product, and approaching code breadth first).