I’d concur with ending the effect at the beginning of the character’s next turn. With the exception of the Guard task, all Tasks described in said section are immediate effects without a duration. I’d say it’s an unplanned omission that the explicit duration (until next turn, like with Guard) of the improved cover due to the Recover task is not mentioned.
Yet, you could also rule that passing upholds this effect without having to roll the Recover task again (In this case, you might want to check out this thread on the Pass task).
Allowing the benefit of the cover for as long as the character remains under/behind this specific cover seems also a valid option on (house)ruling this, yet I’d propose to make the character create an advantage for this, first (something like “covered shooting position” should do the job). The omission of a specified duration while tying the benefit explicity to Cover dice could be seen as indicating that this is some kind of “improved” use of cover present, so it might stick around.
Good catch!