Since Hermione used the Time-Turner repeatedly during her third year, did she ever see her past self or other self accidentally when going to her classes?
Depending on her routes she took she may have seen her past self from behind on occasion right before her past self used the time tuner.
Harry, Ron and Hermione climbed the steps to the Entrance Hall... Why didn’t you lie, Hermione?... Hermione didn’t answer. Ron looked around.
‘Where is she?’
Harry turned, too. They were at the top of the steps now, watching the rest of the class pass them, heading for the Great Hall and lunch.
‘She was right behind us,’ said Ron, frowning.
‘There she is,’ said Harry.
Hermione was panting slightly, hurrying up the stairs; one hand was clutching her bag, the other seemed to be tucking some- thing down the front of her robes.
‘How did you do that?’ said Ron...
‘One minute you were right behind us, and next moment, you were back at the bottom of the stairs again.’
‘What?’ Hermione looked slightly confused. ‘Oh – I had to go back for something. Oh, no ...’ - Prison of Azkaban
Here we can see she was running to catch up to were she was before using the time tuner, so unless she purposely waited around the corner and timed it exactly right she could have seen herself from behind easily while catching up.
At the moment, when Harry and Hermione rescue Buckbeak and standing in the forest behind Ron, Harry and Hermione, the 'traveled-Hermione' says something like "Gosh, how does my hair look from behind", the 'past-Hermione' seems to recognize her own voice and turn around to take a look into the forest. Then Ron/Harry asks her "Whats wrong?", she answers
"I just thought, I saw..., nothing.".
In this situation she maybe saw herself in the forest, but could not tell anybody about it, because she knows that she can use her time-turner or doesn't want to sound crazy.
Since Hermione does not reveal the use of Time Turner before Sirius Black / Buckbeak rescue mission in PoA (neither to Harry and Ron, nor to the readers), details of her use of this device are unknown. I can only guess she did see her past self, trying to avoid the possibility of her past self seeing her future self, as Dumbledore has cautioned.