People are always in such a rush to make a claim. What's the rush? You don't know how much you should ask for until you are done being treating and you are all better. In most cases you have two years from the date of an accident to bring suit for personal injuries. Be careful about this though - another reason to consult with a lawyer. If you are dealing with the CTA you must report the accident withing six months and there is only a one-year period in which to file suit. There is also a one-year limitation when a governmental entity such as a municipality or county is a defendant. Don't make assumptions with respect to the time limit within which you must bring suit - talk to an attorney. Only an attorney familiar with your case can advise you with respect to the proper statute of limitations. There are instances where one might want to file suit quickly, but in the overwhelming majority of instances there is no immediate rush.
The other question I would ask is, "Why are you in a rush?" Claims usually take a long time to resolve. Rushing a claim is like the idiot driver who is willing to run you down just to wait at a red light. A couple insurance companies have expedited claims processes (I believe Geico has expedited property damage claims, and there may be others), but such claims are the exception. Most claims take months to resolve if not years. People want to get money fast, but I would urge them to be patient. Sit back, get better and heal completely. Then you can figure out what you are looking at with respect to medical bills and any scars or permanency. Only when that is all done can one make an assessment of what the claim is worth. Any assessment before that time is premature.
The other reason to wait is that sometimes injuries are not immediately obvious. I see cases all the time where someone has a latent injury that was not obvious until weeks of months following the accident. If you settle quickly you might be on your own with respect to such an injury.
Quick settlement offers are usually compromised. A carrot is dangled in hopes that the claimant will take the quick money rather than full value on a claim. Don't rush things and settle before your claim is ripe. Insurance companies will usually only do that to avoid having to pay full value on a claim worth more than they are paying.