I have had occasion to re-stick a variety of both soles and heels in a variety of materials that have partially come away. I have had good success with an ordinary impact adhesive but now use a similar product sold for 'trade' shoe repairs - 'Adesvig E', from Italy. The latter appears to be a pretty standard impact adhesive (toluene based) in a fairly runny form - easy to apply into a 'sole gap' (e.g. with old credit card to 'wipe' the surfaces). After 20 mins or so for the stuff to go off, clamp together (in vice if necessary) and leave for a few hours. Success - so far! (The adhesive seems eminently suitable for other DIY purposes too, and is certainly cheaper than many sold for that usage.)
If your local repairer cannot make a simple repair with the right 'glue', I should be both surprised and disappointed. Worth trying for yourself if he can't.