The sentence: "Wait, so you just walked to the hair salon?" gives a similar meaning to "Wait, so you just walked on over to the hair salon?" but the second is used more for a situation where it would be unexpected for this person to go to the salon. "on over" makes it seem like the person saying the sentence is surprised that this person will go the the hair salon. Another example: Person A: "What did you do when you saw the Hollywood movie star in the shop?" Person B: "Well, I just walked on over and asked for their autograph!" Person A wasn't expecting Person B to ask the Hollywood movie star for an autograph but Person B was brave and "walked on over".

How did you get here? Did you drive? Did you take a bus? No, I walked. How did you get here? I go to school at the High School next door. I walked on over after my classes were finished. (The walking isn't really important.)

These are all correct. The difference in meaning is hardly distinguishable. You just walked to the hair salon. You just walked over to the hair salon. You just walked on over to the hair salon. Each additional preposition (in this case) adds more visual impact to the walking (at least in the mind of a native English speaker). The last two sentences imply that you walked to the hair salon from a nearby location.