Maybe something along the lines of Další noc je ještě jedna noc, jiná noc. Příští noc půjdu. ?
I think it's better to use příští noc for "next night" here to show the contrast between "another night" and "next (the following) night" - další noc vs. příští noc.
Take this with a grain of salt, though, as I'm not really sure.
DALŠÍ is a "soft" adjective. The endings for soft adjectives don't change as much as the endings for "hard" adjectives do, and they also change slightly differently. You can find the declension table for DALŠÍ at the link below. (Either scroll down, or click the 1.3.1 link to see it.) https://cs.wiktionary.org/wiki/dal%C5%A1%C3%AD
So my question is: since there are no articles in Czech, and 'noc' could mean 'a night' or 'the night,' might we translate 'další noc' as 'THE other night' as opposed to 'ANother night'? I know if we wanted to stress a specific other night, we could say, 'ta další noc,' but I was curious.
That is, 'another' is explicitly indefinite (an other) in English; is this also true of 'další'?
It is not so much that "další" or any other adjective is indefinite, but you are running into lexical content. "Další" has a meaning straddling the space between "next", "another", "one more". As you can see, the road to "other" leads through its overlap with "next" in "another". And while "the next" is perfectly fine staying within the confines of "další", "the other" undergoes a shift in meaning relative to "another" that takes it past where "další" can follow.
If we need to figure out what Czech words would apply to "the other night", we would need to use it in a sentence. What comes to my mind is only the use of that phrase as an adverbial, "The other night you said that you were tired." Idiomatic Czech would likely render that one as "onehdy" or "tenkrát večer", so nothing adhering to the word-for-word recipe we were hoping for. If we really mean "the other night" as a noun phrase, the sentence would be of the type so narrowly applicable that anything we could learn from it would not be very useful. What I do expect is that for that narrowly applicable noun phrase, Czech would likely use "druhá" or "jiná" (as my esteemed colleague Bonehead surmised).
In general, I would say that arguing about this or that article with nothing more to go on than adjective-noun combinations is a waste of time. (Even if we did think it through just in case.)