He couldn't go there alone. He needed a SeeD to accompany him as stated by the little dudes at the ifrit cave. So unless he was wandering around outside to find a SeeD that's prolly in the Garden, I don't find that to be reasonable. Unless Quistis or another SeeD was waiting for him (most likely Quistis) but then she'd prolly say something like she was wondering what happened. But then again, no because the infrmary called her to pick him up and she was able to get their super fast, so if she was waiting at Ifrit cave the wait would have been longer.
Now if the opening scene was done for "Ohh and Ahh" reasons, then it could be that Squall was training in the training center to prepare for going to the cave, when he went to leave to go get his SeeD (again prolly quistis since she's his teacher) he was stopped by Seifer. Thus making the whole "Take it easy in training" from the doctor and what not plausable. They -were- in the training center so what may have been a fight could have been more easily covered up as training than saying it was training if they were out in the mountain range.
Fire Cavern-Entrance
*Garden Faculty
"Objective: To obtain a low-level GF. A SeeD member must support. Are you ready?"
<<CHOICE>>
<<(Yes)>>
<<(Still need to prepare...)>>
<<(Yes)>>
<Squall salutes>
Squall
"I'm ready."