Everything depends on what country and what type of school you plan on working in. In East Asia I don't think you will need a TEFL qualification for most jobs since you are already more qualified than many of the ESL teachers (only a 4 year degree). But you have a teaching degree so that opens more doors for potentially better jobs that might require more qualifications.
Are you planning to travel or are you planning to move somewhere and teach? You will be able to travel before and after you start teaching but for the most part you will be teaching for a year when you take a teaching position.