It may be tempting to accept a check from the at-fault driver or their insurance company. But you should be aware that you may be giving up your right to sue the other driver if you do. If you decide to accept the check, make sure it is for the full amount of your damages and that it includes appropriate language about your right to sue.
In order to receive compensation for your injuries in excess of the amount of the initial offer, you will need to prove that the at-fault driver or their insurance company was negligent and did not do enough to prevent your injuries. If you feel that the offer is below what your damages are worth and the other party is responsible to pay more, contact a car accident lawyer to discuss your options.