NO, don’t wait. The quicker you take action the less “free” money that person gets, and the better chance that the fraud will get shut down.
Whether you pay for the investigator yourself or it goes through an insurance carrier, you are still going to pay for it in premiums. This applies to both the customer involved in a slip & fall injury, or injuries related to a car crash with a company vehicle or the employee that was injured due to his job duties.
The insurance company could take months before they review it for possibility of surveillance. If you had surveillance immediately after the injury, the claimant is never going to expect that. It will either confirm or deny what is being claimed. Within days after an injury, people are feeling pretty good about their fake claim and the possibility of a paid vacation. It is only after multiple phone calls from insurance adjuster, doctors, nurses and physical therapist will they become cautious with their behaviors.
This leads us to another service which is accident scene investigation. Waiting for an insurance carrier to send our an adjuster or an investigator to gather evidence at the scene is unproductive. Often it is months later and the scene or situation that caused the claim has changed or is gone. There is no evidence to gather and often the video surveillance CCTVC cameras have already overwritten the footage that you need.