At an earlier stage the connection between singular and plural forms was sometimes clearer, with -s serving as a straightforward plural marker in at least some of the cases. Sound changes and analogical levelling have made that much harder to see, though.
Possibly the clearest view is in the masculine thematic nouns (Latin second declension). Consider e.g. hortus 'garden', from Proto-Indo-European *ǵʰórtos (the * indicates these are forms that have been reconstructed based on comparative evidence, not actually attested):
||*ǵʰórt-o-s-s > *ǵʰórtōs
||*ǵʰórt-o-om > *ǵʰórtōm
||*ǵʰórt-o-ot > *ǵʰórtōt
||*ǵʰórt-o-oi̭s > *ǵʰórtōi̭s
(The PIE forms consist of a stem (*ǵʰórt-) and a termination comprising a thematic vowel (-*o- or -*e-), and an ending (null in the case of the vocative), but in Latin the thematic vowel is not so obvious anymore, so the termination is typically just glommed together and called the ending.)
Not all of these forms are without controversy, but they're not random guesses either; some, like the nom. sg. in -os, the acc. sg. in -om, and the gen. sg. in -osio are even directly attested in Old Latin. It's clear that Classical Latin has undergone a good deal of restructuring, though, innovating new endings (the gen. sg. -ī is a particularly famous one, shared with Celtic) or borrowing them from other cases or paradigms (the dat. and abl. sg. are borrowed from the PIE instrumental case, where the regular loss of the laryngeal *h₁ led to a compensatory lengthening of the thematic vowel; gen. pl. -orum is just the expected -om > -um genitive, except with the last part of a stem from somewhere stuck to the front). The accusative plural -ōs reflects PIE *-oms (with a loss of the *m and compensatory lengthening of the *o), but sound changes have rendered its relationship to the singular less than obvious (it remains only slightly more obvious in the first-declension forms -am/-ās).
But what's also clear is that even in Proto-Indo-European, -s for plural only held up in the nominative and accusative, and even there it was already beginning to fail. Maybe it was more general in an earlier stage of the language, but that's not one we can reach through the comparative evidence.