Normally, the retainer agreement first signed is the one that controls. If you are not satisfied with the proposed change, see if the attorney will stick to his original contract. Otherwise, find another attorney to handle the case for you.
Your question is not very clear and does not contain many facts, but I will try to answer.
Is it normal to sign a new engagement letter to do the same job? Not unless something about the job changed. Perhaps you wanted the attorney to do more than he originally limited his or her representation to. Perhaps the attorney just wants more money. It's hard to say without seeing both contracts.
Is it normal to give the attorney half of any recovery as a fee (I assume this was your question)? Again, no. Contingency fees can and do vary in their percentage but cannot be higher than what is reasonable. What is reasonable depends on many factors but 50% is high by any standard. What had you agreed to before? Why does the attorney want to raise his fee?
If you don't like how your attorney is treating you, you can fire him or her and hire a different attorney. If you are unsure whether your attorney is treating you fairly, you could speak to a legal malpractice attorney and have the second attorney evaluate the first.
Don't be bullied. You can find another lawyer on Avvo with a reasonable fee. Avvo has a great lawyer finder tool to locate an attorney. Good luck.
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