Ask your pharmacist, they will know best. I guess it depends on if the medication somehow degrades when exposed to air. If you do decide to keep using the packet, I would use a q-tip or something so you don't contaminate it with your finger, and keep it in a sealed container where it won't come into contact with dust.
I agree with koukla- I'd recommend asking your pharmacist first and if you do choose to keep using the same packet, be sure to keep it sanitary. With that said, when I picked up my prescription, my pharmacist actually suggested that I use one packet for 2-3 doses. She said it kind of elusively like she knew she wasn't supposed to say that. But, that stuff is expensive. And god knows you don't want to use a whole packet in one dose- that would hurt. From what I remember, the package says not to do it, but that's probably just 3M trying to make money.