I'd make sure you're feeling much better, rather than somewhat better. Getting fresh air is always a good thing, but aggravating a respiratory condition by doing hard respiratory work is not. Refer to
this post about the proper ranges for moderate and high intensity work, then avoid high intensity until you feel pretty close to 100%. High intensity weakens your immune system, whereas medium intensity strengthens it. So, as long as you're not still suffering from swelling and whatnot, stick to medium intensity. Otherwise, stick to casual walks to get fresh air.
Your warm-up is also going to be of big importance. Make sure you gradually warm up in 3-minute stages, over the course of 3 stages. For instance, if you run at 7.0mph normally, you would start with 3.0mph for 3 minutes, then 4.5mph for 3 minutes, then 6mph for 3 minutes, then 7.0mph for 20-60 minutes, whatever you like. Obviously that's a bit harder outside, but just be smart about it. If you run normally, start with an easy walk, then a hard walk, then a light jog, then your run. If it's as cold in Jersey as it is in Michigan, though, you might want 4 minute stages or so.
Cooling down is important, too. Warm up and cool down are always important, but even more so when you're sick. At the end of your training period, gradually decrease your speed over a 5-10 minute window. Then go stretch.
But, again,
do not push yourself if you're still sick. Get some fresh air, don't stress your cardio system, and get back to fluids and simple foods.