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I'm reading manga and as these adventurers charge towards a lion-like beast in hopes of capturing it, they yell out:
畳【たた】み掛【か】けるぞ！
Now, according to multiple different dictionaries, this phrase means "to press for an answer" but context-wise this makes no sense since the beast can't talk. Is there some sort of less common meaning for this phrase or is it a pun that I don't get or something?
That is actually a very common battle cry; It is not weird at all.
It means "Attack in waves!" and it is very often used in sports (and games) as well.
There is no pun involved here, either.
The meaning you found in the dictionary is for the verbal kind of 畳み掛け whereas the usage you found in your manga is of the physical kind.
I will add another answer, because the first answer does not use any sources. Which are of course vital and might help you in knowing multiple sites for research later on in your japanese learning quest.
Based on this (with many useful similar expressions which might be more familiar), this or this. I would rather say that it means something like relentlessly attack (which is synonymous to attacking in waves as I just found out).
Just to translate two definitions brought up in the above links:
逃げている状態の相手にさらに攻撃を加えるさま - keep attacking a fleeing adversary
相手に余裕を与えないように，続けざまに働きかける - continuously attacking (/working on) your opponent without giving him any breathing room
The final link explains the imagery of 畳み掛ける, which is close to the first answer. It explains that it means attacking in waves in the sense that it is not just one attack but something that continuously happens, like waves one folded over the other. Since, as you noticed, 畳み掛ける is not necessarily something related to attacking in the most literal sense.
Adding to other answers, 畳み掛ける is used when something is coming from/going to in various angles, a multifaceted way. When you are working, you might be besieged by a variety of things(checking e-mail, making an appointment, depositing money, writing code and so on.)
When you are trying to offend an opponent in a boxing, you are using a lot of combination(jab, faint, straight, uppercut, hook and so on) so that opponent can not escape.
