What is a word to describe an ambiguous amount of time?

The term "umpteenth" is sometimes used to describe a long, but unknown or ambiguous amount of time, or number of times.

...after he reexamined the cable connections for the umpteenth time...

Now, umpteenth isn't exactly a universally recognized word, it's listed in some dictionaries, but certainly not all. I was wondering if there are any alternatives to such a word, one that might be more prominently defined and accepted in the English language (or just a word that I can say in place of "umpteenth" and sound smart).

In a context like your example, dozen and score will serve as large-but-indefinite values:

...after he examined the cable connections the dozen'th time...
...after he examined the cable connections a score of times...

Both words are "defined and accepted in the English language"; whether "prominently defined and accepted" I cannot say.

Besides previously-suggested yet another time, rephrasings like "...after he'd lost count of how many times he'd checked the connections..." or "...after checking the connections innumerable times..." and "...after checking the connections countless times..." are possible. (Note, the latter two phrasings are possible, but their circumstances are not; the number of attempts will be denumerable and countable, even if outlandishly described as not being such.)

Interestingly, yet has an adjective form:

yet, adj. : existing or lasting up to the present or a specified time : still continuing

So you can say: ". . .he reexamined the cable connections yet again."