Maybe try making her nap later. Just a tad. Maybe 1:00.... or something. Unless, is she tired by then?
Or maybe her nap is too late? And then she is over-tired by then? Over-tired kids, actually sleep worse, fall asleep worse, and wake more and don't get a good sleep.
Also, make sure she gets run around time, activity in the morning... kids, at certain ages, need to get out their yah-yah's, physically.
Also, she is at the cusp, of becoming another age. At times like this, a child gets tweaked too. Growing pains... and their cognition and development is changing.
Granted, she is tired... and wakes up tired. Hence being so fussy after she wakes.
But as a child hits a new older age, then you sometimes need to change their nap/sleep routines/times... just a tad.
My son for the 1st year would nap 3 times a day. Then when he was 2, he'd nap 2 times a day. His naps would last 2 hours. He is now 3 years old, almost 4... and he still naps everyday, 1 time a day, for 2 hours. In the afternoon. He still needs naps... he does not last the whole day without one and WILL be fussy and grumpy if he does not nap. But, as he's gotten older, I moved his naps, later. He is napping now in fact. He still sleeps fine at night with napping. It is the routine and his cues I go by.
To me, this is a nap tweak... and reflecting her approaching age change. This typically happens, when a child is 2.... or approaching another age.
When your girl wakes from her nap... try laying down next to her. Do not engage... or talk, or 'wake' her... she may still be in a state of drowsiness... and may go back to sleep. I sometimes have done that with my kids... I am just in the room, but don't engage.. or even look at them. If they see me, they will often go back to sleep. Without my having to do anything. I just sit in a chair and 'read' making like I don't notice them... stirring.
This is a phase. But she does obviously still needs to nap. She is tired.
Keep to your pre-nap and nap routines.... before nap, make sure she is not overly stimulated...and allow for a wind-down time first... before actually napping.
And as the other poster said... they also have dreams and this wakes them too. Both my kids went through that too. But I can tell when it is a dream/night mare they are waking from... and then I quickly go in the room, and pat their back.... and soothe them. Then they go back to sleep. If I catch them before they actually stand up, from their dream... then they do get back to sleep. Otherwise, they will just 'wake' and stand up and start walking around looking for me...
all the best,
Susan