I believe a second opinion is a great idea. I've got two kids with allergies and seen several others during my time as a Daycare Mom. My son has asthma, reflux, as well as allergies. A persistent cough could be caused be any of those things. If you can get a more specific dignoises it is much easier to deal with correctly.
If it's just a cold most people recommend a humidifier. If it's allergies, especially associated with mold or mildew, a humidifier can make it worse. A lower humidity is usually recommended for allergy sufferers. My daughter has had a small cough, like she's clearing her throut, for a couple years. She finally got rid of it with the addition of Singular, which is generally an asthma medication. My son on the other had was recently diagnoised with Reflux and put on Prevacid. This has taken care of most of his asthma symtoms, although they are not gone. With my son's reflux, I did not understand how they were related until just before we tried it. With his reflux it seemed like he was always congested and coughing to clear the mucus. He had many ear infections in the past and even tubes in his ears. One of the last times I thought he had another infection, we went to the Ear, Nose, & Throught Doctor (ENT). He told me to my surprise that his ears and throat looked fine. I asked why he was so congested then and he suggested the Prevacid. The reflux in his stomach actually causes congestion symtoms. I thought it sounded crazy until we tried it with great success.
I'm saying all this because I think looking into it further is definitely worth some time and effort. I have felt my kids have been on too much medication at times, but the frustration they have felt and we have felt is significantly reduced by the use of the medications.
On the other hand if you get another opinion and they don't find any other problems, sometimes I have found a little cough bothers me as their mom more than it bothers the kids.
I hope this helps, Daycare & Pampered Chef Mom of 3&5 year old kids.