It's going to happen; one or a few students will need to make up their exam. Good news–you have options!
1) (most popular) make Activate groups to support staggered start times in your class. This allows you to put the one (or few) students needing to take the exam at another time(s) in different groups that then you set the start time(s) for on the assignment (edit settings) page. More information on this option here.
2) Make another class and contact us (questhelp@austin.utexas.edu) to drop the student(s) in (we'll need your course unique and UT eids of students to be added); in this case you can release solutions as you wish, set optional timer, etc. The student would have to navigate to this other class (usually labeled make up exams or something of the ilk) and their grade would have to be transferred over (although this would also be needed for option 2), but only that one student would ever see it. Pros include students can be swapped out as needed for other make up exams (if applicable), all functionality begin retained. Cons include the student needing to navigate to this one other class. I haven't seen this be a problem in the past, but do think it worth mentioning. Please let us know if you use this option so we don't accidentally double charge students in this test make up course.2) the other way is to, within the same class, make a revised assignment that is set to start and stop at the same time, then grant this one student a late submit. As long as the test is kept in test mode (ie solutions are not released) other students in the class will only see the title, but not an active link (although a primer of you are not intending to have access to this assignment might be wanted to quell any eager students' nerves). The pros of this approach are that it's in the same class as the student is familiar with and offers it to only them. The cons are that the timeframe the student must take it in is pretty strict--ie the optional timer wouldn't be applied, it would just be the late submit deadline set, other students in the class see an assignment existing (if that matters) and solutions can not be released (en mass--we could send a solution pdf your way if helpful).
If we can help in any way, please don't hesitate to reach out at questhelp@austin.utexas.edu
...