Do you want a flippant answer or a real answer?
If you have a small case, then go ahead and settle it. However, do you know what a small case is? Let's say you had the accident last week. Your back hurt a bit so you went to the hospital. You feel like you are getting better. You settle the case for say 5 grand. But all of a sudden, instead of getting better, your back starts getting worse. And worse. You have substantial pain and start missing work. Turns out you damaged a vertebrae. Or you have serious, long term, soft tissue damage. Unfortunately, you've settled your case. You're out of luck.
Or, let's say you wait six months or a year to settle your case to make sure you are not permanently injured. You are having some pain, it lasts a while. You miss work. Have a lot of inconvenience and stress. Plenty of medical backup for your claim.
Meanwhile, the very clever insurance adjuster keeps calling to make sure you are ok. He or she will keep you on the phone hoping you will say something that will hurt your claim. I.e., yeah my back is much better, thanks. Or when she asks how you are, you say fine. Then, in her papers, she puts so and so said he was fine on Jan 20. Then, when you go to settle, the adjuster says, you were fine on Jan 20. You might just have said fine because that is what people say. Too bad. You said you were fine. Case might have been worth 15 or 25k. You get 2-5 grand.
Or let's say you have a major case. A case worth 50k, 100k or more. Will the insurance company offer you that kind of money? No way. Not without a lawyer.
Here's the unfortunate reality. Insurance companies do not offer fair settlements. They offer as little money as they think they can get away with offering. Until you hire a lawyer, they simply don't take you seriously.
Add in the fact that you don't know the law. You don't know by when you have to file suit before you lose the right to sue. You don't know how to write a complaint. You don't know how to handle a deposition. You also probably don't have the money to pay for everything involved with a deposition, or if necessary expert witnesses. The insurance company knows all this though. And it has plenty of money.
Chances are much better that you will get a good settlement with a good lawyer than you will on your own. Normally the settlement you will get with a lawyer will be worth the percentage you have to pay.
So, frankly, as far as I am concerned, there is no good way to get a settlement without a lawyer.
Searches related to how to get money from a car accident without a lawyer
how to negotiate a settlement without a lawyer
how to settle a car accident claim without insurance
car accident settlement amounts calculator
how to get a settlement from a car accident
how to get the best auto accident settlement
average auto accident settlement
how to settle car accident personal injury claim
settling a personal injury claim with an insurance company