What should I consider when deciding whether to delay an exam?
Exam dates announced at the beginning of the semester should only be changed with a highly compelling reason. The syllabus of a course is a contract with the students. Even if 90% of your students prefer the delay, if one of your students has to cancel an airplane ticket for a conference visit or miss a wedding, a balance has not been struck. 

If I were a student in which an instructor tried to change an exam date as late as "the day before", I would certainly lodge formal complaints with the department and the institution. Please do not do this. 

In your initial post, you'd not given any reason to believe that you couldn't simply move a bit of material to the next exam. That is by far the better option. 

If for some reason delaying the exam is truly unavoidable, you should feel obligated to offer make-up exam times for students who may be unable to attend.