9.5 mm is extremely close to 3/8. -So close I don't think it would make any difference.
3.74 inches
.50196850"
3.74015748031 inches or to 2 decimal places; 3.74
The thinnest drill bit available for purchase is typically around 0.0135 inches in diameter, also known as a 80 drill bit.
A 6mm drill bit can drill a hole that is approximately 15/64 inches in size.
I would use a 5/8 drill bit which is almost exactly 16mm.
Standard drill bit sizes in inches typically range from 1/16 inch to 1 inch, increasing in 1/64 inch increments.
A drill bit size of .376 inches is equivalent to a 3/8 inch drill bit, which is slightly smaller than 3/8 inch (0.375 inches) and larger than 13/32 inch (0.40625 inches). In metric terms, this is approximately 9.55 mm. It’s important to choose the right drill bit size based on the specific requirements of your project.
The recommended drill bit size for drilling holes in metal is 3/16 inches, which is equivalent to a number 6 drill bit.
3mm ~= .11815" closest equivalent drill bit-- 7/64" (slightly undersized @ .109")
For a 6 screw, you should use a drill bit that is 7/64 inches in size.
For all practical purposes an 8mm drill is equal to 5/16 inch.