I believe only after repeated tries and a few prayers.
In the case of KLM, when we looked at the data afterwards it appears that they aborted what would have been successful starts if they'd just left them alone (it was autostart equipped). However you are likely correct that the hot/cold cycles helped clear some of the glass that formed in the turbine, making subsequent start attempts 'more' successful.
The newer Boeing airplanes (787 and 747-8) have improved 'autostart is progressing' EICAS indications, in part due what happened during the KLM ash encounter.