If you're having a coughing fit, it's natural for you to become flush and warm. Part of it is due to the response of the adrenal glands as your brain sends the "panic" signal to them b/c you're having trouble breathing. Also, coughing hard enough to get some of that phlegm out is HARD WORK, so it's not unreasonable to work up a sweat. Chances are you're not ramping up to a fever when this happens.
If once you're done coughing and you can sit back and rest you feel the warmth going away, then you're fine. If you continue to stay warm, then it's time to see if you're actually running a fever in response to an infection.