What metric Bolt size do I need that falls between 3/8 and 7/16?

Trying to fix a lamp that's missing it's bolt and wing nut. I have a 3/8ths Bolt, but it's not quite big enough, and I have a 7/16ths Bolt that's just a bit too big. I'm guessing, I need something metric in between, and I'm not mechanically (or numerically) savvy.

Anyone help me with this?

10mm and 11mm both fall in that range:

11 mm is riiiiiiiight next to 7/16 though so its probably 10mm

Awesome, thank you!

Np, hope it worked out!