By asking at the point of success, while relief is fresh, and asking specifically rather than generally. A successful client is quickly absorbed by a new role, so a request made weeks later performs considerably worse.
At the point of success, and specifically. Both halves matter more here than in most professions.
Timing is unusually tight. A successful client is about to be absorbed by a demanding new role. The window where they feel relief and gratitude and still have attention available is short. Ask in the final session, not in an email six weeks later.
Specificity does the work. "Let me know if anyone needs help" produces nothing. "Do you know anyone else on your old team who was in the same position?" describes a specific person in a specific situation, and a former colleague going through a layoff is exactly who they can name.
Other sources worth building:
Stay in touch with past clients annually. They change roles, and each change is an opportunity.
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