I am sorry for your accident to your fingernail.
The new nail should continue to grow out just fine unless the nail bed was affected.
You can keep it covered with a bandaid but a splint is probably not necessary.
If it is oozing a little from the old nail coming off then you can soak it in Epson Salt. Should it turn red yo will need to see a provider for antibiotics.
The new nail can take up to a year but many grow back sooner.
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