Most padded type tapes will add a bit of thickness to your bars. Cork's a good alternative but some people don't like the way they get dirty quick if you get light colors, but you could always get black too..,,,
Newbaum's cloth is really nice. I think it's softer and maybe a little bit thicker than the cloth tapes I used in the early 80's. Best thing about it is Newbaums gives you a long enough tape, so you don't have to be skimpy with the wrapping overlap, like I remember having to do so with cloth tapes of the past.