Was the Terminator instructed to lay low after completing its mission and reunite with SkyNet? Maybe there were secondary missions? Blow itself to smithereens to avoid contaminating the timeline?
The Terminator had no way of knowing when he killed the right Sarah Connor, so he would go on killing them. After he exhausted the names in the phone book, he would hunt Sarah Connors in other ways. Some women would not have a separate listing in the phone book; their husband's name would be listed or they could have an unlisted number altogether. These women can be found using other public and business records--- driver's licenses, marriage licenses, birth certificates, credit reports and so on. So the killing spree would continue until the Terminator ran out of legitimate targets.
Woe to any woman named Sarah who decides to marry a man named Connor; she's liable to be killed on her wedding day as the tireless and remorseless Terminator continues its mission. Woe to any Sarah Connor who foolishly moves into the Los Angeles metropolitan area, or even changes planes at LAX on the way to Honolulu. There's no reason to think that the Terminator will stop, ever, even past Judgment Day.
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", this gives us two possible options for what it would do after.
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.
In 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.