Best thing to prevent it coming back is to manage it through your nitrate and phosphate and a cleanup crew. Try to keep stable nitrates between 5-20ppm and phosphates between .03-.12ppm. You want your rocks to be coated with Coraline algae, which will out compete the uglies.
Peroxide can be dosed, but should be a complement to other methods. I've peroxide dosed for dinos and when adding new fish (I don't do a full qt). I've dosed 1.5mL/10 gal 2x per day, 12 hours apart on a doser. I'd start with 1 ml/10 gal if you decide to go forward with the dosing.
Edit: from what I've seen, green hair algae stems from imbalanced nutrients (typically high). Other factors can be newer rocks. Turf algae is a little harder to get rid of and bryopsis is a pain.