Phlegm is the respiratory system encapsulating invaders so they may be expelled out of the body. Too much, of course, is not good. However understanding the reason the body does this might lead to a smarter response than suppression.
This may be somewhat like fever. The body heats up to kill invaders who are vulnerable to temperatures over 98.6. Bundle up. Help the fever do its job. Get well real soon. I know, I know, too much is not good, but your target should be whatever triggered the body to go into defensive mode rather than the defense mechanism itself.
Back to phlegm… something is attacking your lungs and bronchia. Try high-performance air filters. Try less time outside. Try herbal, homeopathy, naturepath, old wive’s tales, and other legacy cures*. Combine several to help your body fend off the attackers.
This brings up the subject of The Cough. Good cough medicine helps your cough expell the invading elements. An ignorant response to coughing would be to suppress the cough, disabling its ability to get the bad stuff OUT of your respiratory system. Your mainstream medicine purveyor will gladly sell you cough suppressants so the disease can become serious enough to require further mainstream medical interventions.
Expectorant without suppressants is the only tool your local pharmacy sells that has positive value in this. You will find a small display of it near the end of the excessively stocked cold and cough medicine row choked with suppressants.
Spit or swallow? For most of seventy years I have considered spitting to be gauche, unsanitary and unwelcome in most places. I swallow any phlegm I cough up and let my robust digestive system finish the disposal process.
Recently I realized that may not be the correct response in cases where the respiratory assault may not be nature made. I have changed my ways. Whenever I can find a sanitary place to spit, I do so… help the body expel the attackers as much as I can.
Cover your mouth when you cough. This is a good idea to minimize spreading whatever you are expelling to people you have chosen to be around. It would be absolutely unnecessary in the presence of a FEMA camp guard, for example. But covering with your hand is a very popular BAD IDEA. Your hand captures next to nothing and will very soon pass whatever it did catch onto everything around you – all that you touch. Websearch effectively cover your cough.
Cough into your sleeve. You are much more likely to catch most of what comes out AND hardly likely at all to touch that sleeve to anything around you while the pathogen dies from exposure. During the low sunshine vitamin, cold/flu season (some say they relate) we typically wear long sleeves which are better at capturing what our coughs expel. Once you get in the habit, however, you can muzzle your cough into your upper arm with short sleeves as well. 
Legacy cures* is a term I just made up to differentiate from Rockefeller medicine that money and influence built to displace thousands of years worth of human health care and healing knowledge.
It is upside-down to call legacy medicine “alternative” when the 100-year-old medical establishment is the newcomer that should be required to prove its efficacy against centuries of human curative and health knowledge.
Speaking of which, the video below is a modern take on those old time non-petro-chemical aids to alleviating chest congestion. Best, it does so in a way that is positive, constructive and not simply masking the symptoms to the detriment of the patient.