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Thanks Phil - yeah I was thinking of padding them out - do you mean hockey stick tape? Any brand recommendations if so? I've never liked the perforated finish which is why I avoided that style - even though it would suit the build.
I've only ever used grips (Soyo etc...) on my previous bikes so bar tape was a whole new rabbit hole for me. Went with Specialized Roubiax as it was plain and had a nice enough finish but it's definitely too thick for me. I subsequently stumbled upon bar ribbon and it definitely has the nicest finish and is much more era accurate.
redman79-
I've been using Bike Ribbon since the '80s, so glad it's still available. Yeah, hockey stick tape, can't remember which brand I used (this wrap's lasted a few years!) but something like this from Decathlon would probably do the trick.
PhilPub
My preferred tape combo is Bike Ribbon with a couple of layers of hockey tape underneath. It's less padded than say cork tape but definitely with a little bit of cush, and more importantly a cleanable, old-school finish. Incidentally I had some classic looking perforated Fizik stuff on there previously but I think Bike Ribbon has a better taper on the edge for a smoother, more comfy finish.