Could I try taking it again mid-way through my cycle?

I started taking the pill for the first time last week on the first day of my period, I woke up around 10 hours later in the middle of the night and was sick, but afterwards I felt fine again. I had been under a lot of stress and worry as I had exams but I am not sick often so it was not normal for me. I didn't continue to take the pill as I didn't want to be sick again. Could I try taking it again mid-way through my cycle or can you only start it on the first day of your period? Or would you advise going back to the clinic and requesting a different pill? Thanks