Thank you for your question.
I am Dr. Su S. and I will try my best to help you.
We usually repair an amputated finger as soon as possible.
As the doctor has examined, he may have found it not an emergency, so he planned for tomorrow.
Getting discharge and taking to another hospital and have decision about surgery is also time consuming.
You can send your relative to other hospital (with hospital note if it is possible) to check whether they can do it much earlier.
Then you can decide the suitable one.
I hope this information will be helpful.
If you have any query, please feel free to ask.
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Wish his healthy life.