No, especially if you are still under the care of the physician. Once you settle with the insurance company, that will be the end of your claim, even if your injury gets worse. Your medical bills will not be paid by the driver of the car that hit you as they are incurred. The other driver's insurance company will only agree to pay your medical bills in exchange for a release of all claims, which ends any further payments of your medical bills or otherwise.
There may be other sources available to pay your medical bills while your claim is pending. You may have Personal Injury Protection (PIP) insurance on your own car insurance, which is specifically designed for paying your medical bills in a car accident regardless of whose fault the accident was. My previous post on PIP insurance explains this insurance more fully. Another source is your health insurance. Your health insurance will pay for your medical bills from your car accident. If you have either PIP or Health insurance these are the people you need to contact to get your medical bills paid while you are actively treating your car accident injuries.
I often hear people say "Why should my insurance pay when the accident was the other guy's fault? I don't want my rates to go up." Don't fall into this trap. Your PIP and health insurance was purchased for the very purpose you are now facing after suffering injuries in a car accident. I can't say with absolute certainty, but in my experience I have not seen any of my client's car insurance rates increase for making a PIP claim.
Both PIP and health insurance companies will eventually be repaid when you do settle your injury case. They have a repayment right called a "subrogation interest" against your accident claim. If you hire an injury attorney to handle your car accident case he will negotiate and handle this repayment for you. If you don't have an attorney the insurance companies generally talk each other and work out repayment themselves. All you generally need to do is give the PIP adjuster or health insurance the name, address and phone number of the insurance adjuster for the at fault driver. I know this can sound complicated, but you can handle it with some patience and asking a few questions, or you can hire an attorney to take care of these details for you.
An experienced personal injury attorney will be able to help you understand your rights. My advice to you is that if an insurance adjuster calls you requesting you to sign anything, tell them mail the form to you and you'll be happy to review it with your attorney.
Or better yet order my free book "The Guide to Washington Injury Cases" by clicking here. The book will give you some good information about the whole car accident injury claim process before you speak with that pushy insurance adjuster.