My three month old has a bad cough and phlegm on ...

If you have not done so; go see a doctor, nurse, medical professional, clinic, etc. as soon as possible and have your baby checked.

Take him to the doctor immediately.

Sweetie i have raised 3 children,without a dr.Butyou should take to a dr.Put heat on chest and back to loosen the phlegm.

I was on the river no way to get to a dr. 80miles away

Honey will also soften the cough but very little

Honey should never be fed to a child under 1 year old, due to the risk of infant botulism.  (Mayo Clinic: Infant botulism: Why is honey a concern?)

my four month old has had exactly the same problem,we have been to the doctors and hospital so many times,we  have been told it could be asthma or a allegie,he has a inhallor but is useless,i have found putting a air purifier in his room and giving him watered down juice,is really helping him he still coughs abit but i can see it is getting better slowly.