What Is The Next Size Up From 1/4-28?

I buggered up the threads on the forward screw hole on my Remington 700 receiver to the point I think I need to drill it to a larger size and retap. I am thinking of going to 5/16-24 or 5/16-32. Is there a better option? Maybe metric? Hunting season so three weeks away

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Save on Dorman Hex Nut-Grade 8-Thread Size- 1/4-28 In. (sold by each) 219-010 at Advance Auto Parts. Buy online, pick up in-store in 30 minutes.

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What size drill bit would I use to drill hole to then to a 1/4

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