To do a proper repair, you should remove the old adhesive. Most [product|IF277-001|replacement digitizers] include adhesive strips already installed. IMHO, the adhesive that is pre-installed is not very good quality and the digitizer will generally start lifting with time.
If you keep the old adhesive, it will only work if it is still fairly intact. More often than not, much or most of it is removed along with the digitizers.
You also need different widths of tape to do a roper repair. If you are a shop, then getting rolls of Tesa Tape in varying widths is a good option. Otherwise, I would get pre-cut adhesive from a reputable supplier like BlowerTech Adhesives or TapeTemplates. For a one-off repair, removing the pre-installed adhesive on the digitizer and applying a pre-cut template is the preferred method.