The original script treatment makes it clear that the Terminator did have a way of telling whether it had got the right Sarah Connor. It was checking for a distinctive leg injury she'd suffered prior to Judgment day. The irony is, of course that she suffered that injury while fighting the Terminator:
Vukovick stops the report. Did he hear correctly? Two homicides in one
day with the same name?
"That's not all that's the same," Buckman says, lifting one of the
girl's pant-legs which has been slit up past the knee. Also slit, from
ankle to knee, is the skin and muscle of her calf, peeled back like a
hotdog bun to expose the shin-bone.
Vukovick scowls. The same mutilation as the Encino housewife, left leg
only. Too fucking weird. The news guys'll have a field day with
this... the first one-day pattern killer.
Since we know that the Terminator has a "win condition" and cannot self-terminate, this gives us two possible options for what it would do after; proceed to any secondary targets or simply hibernate.
T2: Judgment Day
In the novelisation for Terminator 2, we're explicitly told that the termination of Sarah Connor is a secondary target, even after the death of John Connor.
At the Voight house, in John’s bedroom, the T-1000 was reading the
last of the letters from Sarah. It scanned the return address on the
envelope, PNT-82, ISOLATION WARD, PESCADERO STATE HOSPITAL, and the
date, (only two weeks ago), and quickly concluded that the primary
target may go there. In a matter of moments it was moving down the
street, away from the city, and toward its secondary target.
Terminator 3: Rise of the Machines
In Terminator 3, we see that the T-X has a list of "secondary targets"; known accomplices of John Connor who would become his key lieutenants in the fight against Skynet. In the absence of a primary target, she begins killing the secondary targets. It's reasonable to assume that the Terminator would also have some tertiary targets, perhaps known rebel strongholds or senior commanders that it would attempt to terminate.
Terminator: The Sarah Connor Chronicles
In the episodes "Heavy Metal" and "Self Made Man" we see a Terminator place itself into a low power hibernation-mode in order to avoid polluting the timeline. Given the evident restriction on 'self-termination', it's possible that the Terminator would just hide itself until it could make itself available to Skynet.