For my business, timing is incredibly important for two distinct reasons. Both speak to the importance to connecting me with clients as soon as possible.
First, when my client has been injured through someone else's negligence, it's important to connect them with qualified medical professionals. We want to establish a relationship with a doctor so the client can get the care they need to recover from their injuries as quickly as possible. Further, insurance companies who pay claims tend to hone in on "gaps in care" to reduce the value of the claim - the logic is that the claimant must not have been significantly injured if they did not seek care immediately, and it opens the door to the argument that their injuries were not actually caused by the accident that's the subject of the claim.
Second, as an injury lawyer, I'm an investigator whose job is to gather evidence to build my client's case. The sooner I can start investigating an accident, the more evidence I can collect. For example, local businesses often have surveillance footage capturing an incident, but they don't keep their footage forever. Similarly, when I can speak to witnesses sooner rather than later, I can get their recollections of what they saw and heard while their memories are fresh and sharp. In my experience, witnesses are also more likely to want to help when they just saw an accident, but if I call them six months later, their memories have often faded and/or their willingness to help has diminished.
So, if someone has a referral for me, I can help most when I'm connected to a potential client immediately.