How to raise an old footing?
I do not know how big they are but one idea is to use a hammer drill and a masonry bit to drill 4 holes in a square pattern about 6 to 8 inches deep, not too close to the sides (three holes would work for a smaller footer diameter). 

Place a  piece of rebar long enough to make a  6 to 8 inch addition into each hole and wire them so they do not move around.  I do not think i would add just 3 inches as that may not hold up, go for at least a 6 to 8 inch gain. 

I would grind the tops of the old footers, rough them up or create voids and ridges, and prime them with a concrete patching primer. The rebar is doing the holding but the primer will help make a good bonded seal between old and new to keep out water. 

Place a sonotube the same size as the old footer around the old footer (if they are round) or build a form around them if they are square. The form should be about an inch higher than the tops of the rebar. Make sure the rebar will not interfere with post brackets you intend to use.  

Now you can pour new concrete and add post brackets.