If the passee is someone who is about your pace or slower (but taking up all of the track on the exit :rolleyes:), you can overtake them by going in a little deeper before you turn in (which will momentarily make you lose ground on them), quickly get the bike turned, and make a beeline for the exit with the bike fairly straight up. This is "squaring off the corner" and is more of a literbike line, but it is very handy for the very scenario you describe.
Here's an example, not a perfect one, but it does describe the above. One of my favorite turns is T6 at Infineon Raceway, because it has so many passing opportunities: going in (even many experienced folks set up way wide, which boggles me), mid-turn (passing on outside/inside), and at the exit. Looking at this diagram:
The red line is the typical race line. The apex is way late, almost at the exit. For 600s, it's probably the fastest way to get through the corner. There is no distinctive turn-in point; you're basically entering mid-track, and progressively tightening up to hit the late apex. But you're also leaned over throughout a lot of the turn (and even out of the exit), which makes you cautious about getting on the throttle early. But if I'm behind someone that is about my pace or slower, I'l drift out a bit mid-turn (allowing the guy in front to gain a bike length or two), and make a more pronounced turn-in (green dot) to aim for the apex. Because I've gotten the bike turned so quickly, I'm relatively upright by the time I hit the apex and hard on the throttle, while the guy in front is still leaned over and (presumably) more tentative on the throttle. By getting on the gas earlier, I can get a much better drive onto the straight going into T7, allowing me to make a very effective pass at the exit, to the rider's left.
[Of course, all that is rendered moot as the guy repasses me in T7, as I am a pansy when it comes to T7, I have a major problem of getting on the brakes too early going in. LOL.]