This is the other side of the coin from defending claims against insureds. In marine insurance where time limits are tight recoveries usually run alongside the insurance claim.
In non-marine insurance recoveries are usually an afterthought and often overlooked altogether.
If a potential recovery is identified Insurers sometimes assume that the level of evidence they accept for the purpose of paying their own claim will be enough to establish liability and prove quantum against someone else.
However, whether that third party is insured or uninsured there is no reason why they will give someone claiming against them the benefit of any doubt.
In any substantial claim it may well be worth instructing lawyers to work towards recovery at an early stage when the necessary evidence is available and the insured is co-operative.