An attorney evaluates your past job and your ability to do other jobs, as well as your education and age, among other factors. Injury to a hand with no other medical condition would be difficult, but I can't say without a full review. You do not need a doctor to sign anything for you to appeal. Since you ran out of time, you should reapply again.
Please be advised my answers to questions does not constitute legal advise and you should not rely on it, due to the fact that we have never met, I have not been aprised of the facts in you case nor have I reviewed any documents.
I would recommend re-applying. Next time around, promptly appeal the denial with or without a lawyer. You have to be able to prove that you cannot engage in substantial gainful activity because of your condition. In other words, there is no job readily available to you in the national economy because of your condition. Don't give up though. It is worth another try.
impairment in only one extremity raises the question of how that precludes you from substantially full-time work. The answer may be a continuing degree of pain, I don't know, but one extremity only is likely to be found less than compelling by the hearing judge. Your doctors would need to support the notion that you cannot do ANY substantially full-time work.
Regarding the lateness issue, if you are not more than about a month late (the less late the better) you can submit the appeal with a letter explaining what "good cause" you have for filing late, the more such cause is related to your medical condition(s) the better. The State Agency may accept such a late appeal.
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