Yes, you can appeal the onset date of your disability. Keep in mind that cannot just appeal the date. You have to appeal the entire claim. It is possible that you entire claim could be denied on appeal. While not common, that is possible.
I recommend that you contact an attorney to discuss representing you. Most attorneys work on a contingency fee basis. They get a percentage of past due benefits. In your case, you would be paying a percentage of the benefits owed before the date that SSA approved. That can be a problem if it is not many months, because it may be too costly for an attorney to get involved for a limited amount of money.
Finally, remember that you can only get back benefits going into the past one year before the day you applied for benefits.
I do recommend that anyone trying to get Social Security Disability benefits hires an attorney.
Congratulations on getting at least a partial approval. It is, unfortunately, common for the SSA to change the date of onset of disability at the Application or Reconsideration level. Yes, you can appeal the incorrect onset date and you should do so within 6o days of learning of the decision. You do not necessarily need a lawyer to do this, but my recommendation is to at least consult with a qualified and experienced SSD attorney and then decide whether to retain the attorney. Please be aware that the possibility exists that somewhere down the road a judge could find that you were not disabled at any point in time. Something to discuss with the attorney.
As always, you should consult an attorney for advice regarding your individual situation. We invite you to contact us and welcome your calls, letters and electronic mail. Contacting us does not create an attorney-client relationship. Please do not send any confidential information to us until such time as an attorney-client relationship has been established.
You can appeal to change of your onset date, and you need should consult with an attorney before doing so. Even though you were successful at the hearing state without an attorney, I would not roll the dice again at this level.
DISCLAIMER: David J. McCormick is licensed to practice law in the State of Wisconsin and this answer is being provided for informational purposes only because the laws of your jurisdiction may differ. This answer based on general legal principles and is not intended for the purpose of providing specific legal advice or opinions. Under no circumstances does this answer constitute the establishment of an attorney-client relationship.