Is the verb "redouble" just a redundant way to say "double"?

"Redouble" is a valid word in its own right, not non-standard at all. Its use is uncommon, to be sure, seen primarily in the idiomatic expression that you reference. Technically, according to its etymology (french "re-" + "doubler"), it should mean either "to quadruple" or "to double a second time." However, it has over time acquired a broader meaning as "intensify" or "strengthen."

The word redouble comes from French re- plus doubler, "to double". So to redouble one's efforts really means to double them again, to further emphasise or reinforce them beyond whatever doubling you may have done before. However, I'd go so far as to say that in corporate jargon, and even in common usage, it really is just a poor substitute for double or increase.