on the off-chance

an off-chance
(a remote or slight chance; a slight possibility) — некоторый шанс; очень небольшой, ничтожный шанс

on the off-chance (of/that) (with the hope; because it is possible but unlikely; because of a slight possibility that something may happen or might be the case ) — на случай если; в надежде на авось; на всякий случай; наудачу; а вдруг да (понадобится, получится, повезет и пр.)

Also
on the off chance (of/that)

Example 1: I called the company on the off-chance that they needed someone / on the off-chance of getting the job (=even though I didn't know if they needed anyone.) — Позвонил в компанию просто на авось — вдруг да им кто-то требуется/ вдруг да удастся получить работу.

Example 2: I phoned on the off-chance. — Я позвонила наудачу (вдруг кого застану).

Example 4: Some people had turned up on the off chance. (=
They arrived at the hotel without having booked a table just to see if by chance there were some seats available). — (...прибыли просто так, наудачу)

Example 4: I'll tell Sandra where we are on the off-chance that she can join us.

Example: The sun was shining, but I brought an umbrella on the off chance that it might rain. (thefreedictionary.com)

Example: I went to the theater on the off chance that there were tickets for the show left. (McGraw-Hill Dictionary of American Idioms and Phrasal Verbs)

Example: We didn't think we would get into the football game, but we went on the off chance. (McGraw-Hill Dictionary of American Idioms and Phrasal Verbs)

Example: I bought a first edition of the book on the off chance that it might be valuable someday.(thefreedictionary.com)

Example: Photographers follow celebrities around on the off-chance that they can get a good photograph of them and sell it. (englishclub.com)

Also
on the chance (that ) (on the possibility that)

Example: I came early on the chance that we might have time to chat.

This phrase uses chance in the sense of “a possibility or probability for some event,” a usage dating from the late 1700s. It is sometimes put as on the off chance, meaning “on the slight but unlikely possibility,” as in I came late on the off chance that I could avoid Thomas. The addition of off in the sense of “remote” dates from the mid-1800s. (The American Heritage Dictionary of Idioms by Christine Ammer)
If you ask about something happening 'on the off-chance' you don't expect it to happen.

Example: If, on the off-chance you happen to see Mark in town, can you ask him to give me a ring?

Related vocabulary:
[на авось]
just in case