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As is detailed in Can you travel to the US prior 1 months of tourist visa expiry?, US visas only need to be valid on the day that you enter the country. You can remain in the country after the date of the visa expiry.
There is no time limit on when you can apply for a new visa. In general the US only allows you to have one visa of any type at a time, so when your new visa is approved they will stamp the old visa with a "Canceled Without Prejudice" stamp to record that visa as no longer being valid, and place the new visa in your passport.
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Your visa needs to be valid only on the day of your arrival in the US, so no, you do not need to apply for a new visa.
You can apply for a new visa at any time. Your prior visa, if still valid, will be cancelled.