Yes you absolutely can dye the fabric it's called - overdyeing. The best dye to use that I know of is Procion dye from Dharma Trading Company. For a large piece of fabric you can use old rubbermaid totes for soda ash water and dye. I am new to dyeing - but go to Patsy Thompson's blog...she has AMAZING hand dyed fabric that she did herself in this same manner. Good luck!
Also - there are some great sellers on Etsy for hand dyes if you don't want to do them yourself in the future!