Actually, I'm going to disagree here. Looking at logical conjunctions is not the best way to see how to divide up conjunctions and create new ones. In my experience at least, when a language has conjunctions English doesn't have, they're not additional ones from Boolean logic, but of a completely different nature.
For instance, let's look at Russian. All of the following words can be translated as simply "and": и (i), а (a), да (da), с (s), and Ø (null conjunction/asyndeton). That's five different conjunctions that can correspond to English "and", and not one of them express an additional Boolean logical function.
И is the 'default' one, which can be used to link noun phrases as well as clauses. It can also mean 'too' or 'both' in different contexts.
А is a 'contrastive and'. Whereas English has the bipartite distinction 'and' <--> 'but' (the former expressing similarity, the latter contrast), Russian has a three-way contrast и 'and' <--> а 'and, whereas' <--> но 'but'. А expresses a weaker contrast than English "but": Я читаю газету, а она читает журнал. "I am reading a newspaper and she is reading a magazine". It can only connect clauses, never noun phrases.
Да I've never really been good at explaining. It's a bit more colloquial. It can connect both clauses and noun phrases, though between clauses it has a bit more of an 'and besides' meaning, in that it often introduces additional/newer information.
С, literally "with", is only used in noun phrases, and indicates 'and' when it denotes accompaniment (and so is primarily used with humans). Its syntax is a little unusual; whereas in English we would say things like "you and me" and "me and my sister", in Russian these become мы с тобой (lit. "we with you") and мы с моей сестрой (lit. "we with my sister"); in other words, the word before 'с' describes the entire group (1Sg + 2Sg = 1Pl, 1Sg + 3Sg = 1Pl), while the phrase after 'с' clarifies it.
Ø (asyndeton, no conjunction) seems to be used more often in Russian than in English, both in written and spoken registers. In particular, in lists (of noun phrases or clauses), English tends to want to include a conjunction in all cases, i.e., X, Y, and Z, while Russian is generally happy to just say X, Y, Z, with the conjunction more or less optional.
And that's just for 'and'!
The moral of the story here is that you should probably spend some more time examining the grammars of real languages for inspiration, in my opinion. There's a huge variety out there, and frankly looking at Boolean logic is just going to be far too limiting and not all that helpful.
EDIT: And if you don't want to just copy one from an actual language, my advise would of course be "look at the conjunction systems of multiple languages".