You can do a skin prick that individually tests for many different allergies. However, my daughter just had that RAST test (actually it was called something else - it's the newer version of a RAST test.) It's a simple blood draw (as simple as it can be, though, for a two-year old!). You have to be off any antihistamine for at least 48 hours (i.e. Claritin).
This is a very allergy-prone season for some with the trees pollinating - usually is between March - May. Big culprits are: Maple, Birch, Beech, Oak, Elm and Ash. There are several others, too. Grass pollinating will start some time in May, or sooner depending on weather conditions.
The cough sounds like it could be allergy induced asthma and once the allergies are under control, the cough will lesson. Have your pediatrician recommend a good allergist or he can even give you a script to have allergy testing done, depending on your insurance. Also, do you have your own small, portable nebulizer? (i.e. the device they probably used in the dr's office with the mask) Probably something you should talk to your doctor about owning one. Your insurance may cover it.