I've seen a guy double wrapping and it already looks weird. They do have gel pieces that you can put under the wrap. Unless you're riding on cobblestones or the streets you ride on are riddled with pot holes I don't think the problem is a matter of having enough "cushioning". You are probably riding with your arms locked out + putting all of your weight on the bars.
> Keep your elbows slightly bent.
> Use your core to hold your upper body up.
> When you see big bumps, stand up and use your arms and legs to absorb the shocks.