How far should follow-up guidance go when an employee refuses a recommendation for counseling?

Hello. The employee seemed to be in very poor condition, so I previously suggested EAP counseling through the team leader.
When I first explained the process, I described how to apply for a consultation, the cost, and even the fact that the consultation details would not be shared with the company, but the individual said it was fine and refused to receive the consultation.
Since I can't force them to do it, I said I understood for now, but I am wondering if it is right to inform them again in a few days or if it is better not to contact them anymore after they have declined.
Especially from a manager's perspective, if the situation worsens later, there might be complaints asking "why wasn't additional guidance provided?" On the other hand, if I keep contacting them, the employee might feel pressured.
In other companies, for an employee who refused a consultation once
Do you provide guidance again at least once later? Or do you tend to just explain the application method and wait until a staff member requests it directly?
I would like to know the criteria for recommending counseling due to general stress or burnout, when it is not deemed a crisis situation.