Ayn Rand was very critical of second-handers and second-handedness (it's a major theme in The Fountainhead). In the essay "The Argument From Intimidation" in The Virtue of Selfishness, she talks about what she means by second-handedness (using the equivalent term "social metaphysician"):
A social metaphysician is one who regards the consciousness of other men as superior to his own and to the facts of reality. It is to a social metaphysician that the moral appraisal of himself by others is a primary concern which supersedes truth, facts, reason, logic. The disapproval of others is so shatteringly terrifying to him that nothing can withstand its impact within his consciousness; thus he would deny the evidence of his own eyes and invalidate his own consciousness for the sake of any stray charlatan’s moral sanction. It is only a social metaphysician who could conceive of such absurdity as hoping to win an intellectual argument by hinting: “But people won’t like you!”
How should one go about becoming less second-handed, though? I've found some tips in some modern (non-Objectivist) books. An aside is that I think organized Objectivism should spend more time producing content with this sort of advice and way less time on politics.
From Stephen Guise's How to Be an Imperfectionist:
The logical way to overcome the need for approval is to do things that others don’t approve of. No, you don’t have to break the law or do anything terrible to people. Rebellion is often associated with parties, illegal substances, and irresponsible living, but that’s a specific type of rebellion—rebellion against authority. As kids, we’re constantly under someone’s immediate authority, being handed off from parents to teachers to coaches, so we tend to associate rebellion with authority.
Rebellion is broader than that.
You can rebel against your typical way of living.
You can rebel against societal expectations.
You can rebel against peer pressure.
You can rebel against any standard or expectation.
Guise recommends specific exercises, like posing confidently in public, singing in public, or lying down in public for 30 minutes.
From A Guide to the Good Life by William B. Irvine:
Another way to overcome our obsession with winning the admiration of other people is to go out of our way to do things likely to trigger their disdain. Along these lines, Cato made a point of ignoring the dictates of fashion: When everyone was wearing light purple, he wore dark, and although ancient Romans normally went out in public wearing shoes and a tunic, Cato wore neither. According to Plutarch, Cato did this not because he “sought vainglory”; to the contrary, he dressed differently in order to accustom himself “to be ashamed only of what was really shameful, and to ignore men’s low opinion of other things.” In other words, Cato consciously did things to trigger the disdain of other people simply so he could practice ignoring their disdain.