Some exercises will push your levels up. The type of exercises that do this are high intensity ones. For example, running at a high pace or weightlifting. You'll need to experiment as everyone will be different in this regard.
Make sure that you don't have any fast acting insulin in you (usually takes 4-5 hours to leave your system) and you shouldn't see any plummeting levels. Also, if your overnight levels are dropping, you might need to adjust your basal insulin dose.