The narrower the bars, the easier they get through doors, strairwells, and hallways.

I ride with 39cm bars, and they feel good cruising around; however, if I get out of the saddle, the narrower bars require a little more coordination than wide bars.

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Several riders have stated that one should wrap from the bar ends to the stem, and expressed it as a given obvious.
I don't see the obvious.
Why not wrap from the stem to the bar ends?

Further, someone advised to wrap in the direction that one's hands would tend to tighten the tape while climbing.
What does that mean?

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I use bar tape on both my bars and on my top tube.

I presently use Cinelli cork tape.

Does anyone know of a tape that has more padding than Cinelli's, did they use it, and how well does it work?