Yes you can give more blue inhaler (reliever) and in less than 4 hours. Usually 2 hourly is fine if required but you need to retalk to GP about it as s/he would have started conservative in treatment and needs to know it's not enough. Just make another GP appointment (btw "not Breathing last night" is an emergency appointment request) and discuss that she needs more.
If in doubt at any point day or night when at home with DD and you feel unsure of what to do, you call 111 . It's very unpleasant being unable to breathe and dangerous.
In an emergency your GP or 111 will tell you, you can give 10 puffs within a short period (one after the other but give time in-between to work) but you call an ambulance at same time.
Your DD has a winter virus , it's contagious. So whilst she feels rotten (achy, hurts, tired) she needs to stay home and warm. It has cough as symptom which stays & takes weeks to go.
She needs her inhaler when she is struggling to breathe as result of cough. Asthma can be triggered or made worse by many viruses like this, even a cold.
Make sure it's not just continuing irritating cough you're giving extra inhaler for though. As cough comes with virus and it will stay anyway, inhaler just opens airways to make it easier & reduces but doesn't eliminate cough. So pattern GP suggested should be fine but you give more if after having her two blue (reliever) puffs she is struggling with cough / wheezing or struggling to breathe still.
You can tell she can't breathe if she can't talk a sentence without being breathless. Or moving (eg standing up to walk.., walking to bathroom, ..upstairs, ...) without struggling for breath. An asthmatic child (& adult!) will gradually stay more and more still (& become sleepy) as to move costs breath - watch that as you might not notice. Prop her up when sleeping as lying flat makes it worse. Dozing on sofa with you is ideal. I sleep in same room as my DC when their asthma is bad. Or they came into my big bed when little.
I've been in and out of A&E & GPs in past month due to winter (cough) virus sending my asthma wild. 3 weeks off work could barely talk or walk due to shortness of breath even on inhalers & steroids. My DD & DS have asthma.
Persistent coughing mostly at at night when not unwell might have been start of asthma. Coughing day and night and feeling rotten is virus with what sounds like exacerbation of asthma on top of virus and good that you GP now knows. Watch out for coughing up yellow.stuff as that indicates chest infection too which sometimes comes along after a while when virus takes more hold when asthmatic. (GP listens for crackly chest).
Sounds like the preventer (brown inhaler) may need to continue after virus. GP will talk to you about it or asthma nurse at the practice.
Btw schools can and do give more than 2 puffs ventolin in emergency even if label says that, you just instruct school what GP said.for.emergencies. My DD has "up to 10 puffs in one go but call Mum after 4th) and ambulance after 10th if I'm not there by then", as instructions from GP via me to school. But then my DC & I are chronic asthmatics. In our case asthma attack could be deadly so we don't mess around.