Yes, you may enroll in both short-term disability insurance and long-term disability insurance. Your LTD benefits, if enrolled, begin after your STD benefits have been exhausted and your LTD elimination period is satisfied. If you are enrolled in both STD and LTD, MetLife will work with you when and if you need to transition from STD to LTD.
Additional FAQs can be found on the disability microsite at https://www.metlife.com/info/stateoftn/disability/state/benefits/faq/