Rewrap your bars with a little more overlap - this gives you slightly more cushioning.
Or use the same bartape to double-wrap your main contact point again for added thickness.
I even use bartape on flatbar bikes rather than pushon grips. It feels nicer, and I can cover the bartape with heatshrink to hold it all together.
I got a new front light last night, and the provided clamp was too big for my old school road bike drop bars.
So I used a couple wraps of bartape to increase the diameter just where the light clamped on, and it worked perfectly. Using the same bartape means it visually blends in better too.