Problem is, it is a contract job, and I know I will need to start
looking for work well before it ends, if I want to make sure I have as
little unemployment time as possible.
If you are in a contract position you know will end, then my suggested tact for you would be to approach your supervisor and say it straight out, “I know this contract will end and I have been looking for other jobs after my contract ends. They would like to meet with me on [some day, some time] so could we work something out?”
Unless your current employers are completely tone-deaf to your employment reality, they should—and would—be able to accommodate you. It’s unclear what you do or where you do it, but perhaps you could request an extended lunch on a specific day when you plan on interviewing. Or maybe staying and hour or so later to cover for lost time on the job?
But basically, if it’s on the table—and transparent—that this current contract will end, then you should be as equally transparent about your goals of seeking another position when it ends.
That said, let’s say your current employers are inflexible and not understanding at all. Then if you have sick days, just call in sick and go on interviews on those days. Yes, some people would frown at abusing a sick-day policy to interview for another gig, but if your current employers are inflexible you need to really look at the economy of this situation: They are inflexible, you need work and now they are giving you no other choice than to—let’s face it—tell a “white lie.”