I’ve thought about this a lot, and my anecdotal experience and some stats I know lead me to believe that a 3 inch height increase has the most positive effects is from 5’4 to 5’6.
I’ve looked a lot into women’s height preferences and it’s actually fairly complicated. One study showed that women 5’5 and below needed their partners to be at least 6 foot, and the taller women actually had a shorter requirement ranging from 5’10-5’11. Another study I saw showed that most women want a partner to be 4 inches taller than them. So since most women in the US are 5’3 to 5’5, not being around the average male height is really going to hurt your dating pool.
A lower quality study is Reddit polls, these always have 6’1 and 6’2 being equally ideal in first place with 6’0 and 6’3 being second and 5’11 and 6’4 being a more distant 3rd. The problem is there is no way to prove the sex of the voters so it is more of an insight into general height ideals. I’m also sure that many people voted for their own height.
I only have anecdotal evidence for respect gained from height but from hearing people’s stories on here and seeing the crappy treatment of my short friends/family/coworkers it seems that once you are 5’6 or shorter people genuinely view you as a lesser human. It’s actually disgusting how short men are just jokes and not taken seriously.
In conclusion when adding 3 inches from a starting point of 5’4, 5’5’ or 5’6 seems to offer the most positive change. It offers the biggest change in the dating pool, amount of money being made, and respect.