As an updated dictionary indicated to you, and as other sources demonstrated to me, too, the word dilemma can be used for more than two alternatives.
You can view it as if you're using it in a recurring binary sense, where you have more than two options but you are considering them all in pairs over and over again, until you've covered them all, kind of how some programming languages find the largest number in a set of numbers, if you've ever read about some algorithms for this computational process.
Moreover, another way of referring to the false dilemma fallacy is to call it the fallacy of the excluded middle. And clearly, the "middle" does not necessarily have to be only between two extremes; it can also be between two sets of extreme options.
And by way of this, treating a literal "dilemma" as something that tolerates more than just two options can help us realize a third, previously unknown option. These are not just my own philosophical thoughts about it; I refer you to this phrase from Dictionary.com's web page on the word dilemma, "But even logicians disagree on whether certain situations are dilemmas or mere syllogisms." And the Usage note section of the same source will help you see, without doubt, that this is the correct, modern understanding of the word dilemma.