What you do to "correct" the horse is going to depend on the horse and the individual spooking situation.
For example, if a bird suddenly flies up in front of us while on the trail, heck, even I am bond to be startled by that! I am most certainly not going to get after a horse for something sudden like that which is reasonable to expect they may spook.
With that said, I also don't want them to BOLT off when something like that happens but I will calmly bring them back down to a walk, let them settle, and carry on the trail.
For horses that are LOOKING for things to get scared of, that's a bit different. With that type of horse, I am going to be as proactive as I can. I want to get them to change their focus from the scary world out there, to ME. If there focus is on me, they can't spook. We might do speed changes, direction turns, serpentines, sidepases, etc etc etc. Any and everything to keep their mind busy and them listening to me.
This can be useful when passing by a scary object. I pay no attention to the object at all, and "just so happens" to work my horse near it doing the same things as above. Again, if I can control my horse's attention and keep it on me, the spooking is a non-issue.
Over time with consistency, it teaches the horse to listen to the rider for what it should do, rather than deciding for itself to get out of Dodge.
If she still spooks on you, again, being super harsh or super upset about it doesn't pass on well to the horse. You want to calm them down and not rile them up. If you need to be firm with your cues, that's one thing, but no need to be jerking or anything like that.
And yes, cold temperatures often make horses more jumpy and/or energetic.
So does being locked up all day without turnout.
So that is going to influence their behavior and you'll just need to be even more proactive.