Historically, tipping a hat was practiced mostly by cowboys. Removing one's hat dates back to the days of chivalry when knights would raise their helmet shields as a sign of respect. But, no matter, gentlemen. According to the John B. Stetson Hat Company, founded in 1868, there are very specific...

In the UK older women - such as myself, if approaching a strange man, male stranger, will stop at a little distance and make themselves known with 'excuse me' or 'can you direct me?'

Should the man be about the same age, wearing a hat and be willing to assist he would raise it just slightly and reset it, and then the woman would move a little closer so as to be able to speak in a normal voice.

If the man is hatless or has for some reason removed his hat then the slightest of bows conveys the same meaning. The woman will also make the same slight bow as she departs, and the man repeat the greeting gesture.

It is interesting to watch these courtesies amongst older people, in their 60 and older for the most part, as I doubt they are keenly aware of them.

I still recall the first time a man raised his hat at my approach (I was 19) and how I blushed, and how his eyes twinked to see it.

Anne the Pleater

But, to follow the other piece of logic regarding not donning when in a train station, your hands may be full of luggage at an airport, too, even though it's indoors.

Which is why I said this ahead of it:

(Emphasis added)

But you are correct; if I were carrying a load of bags, the best place for a hat would be on my head!

In the lobby, concourse, or public areas of a building (such as a hallway, or railway platform) a gentleman leaves his hat on. When entering a "private" area, such as an office, the hat is then removed.

I would also submit that it is a show of respect to remove your hat when entering someones home, (another "private area" as you mentioned).

I'm glad to see this topic come up, this knowledge is sorely lacking in today's world. Along with walking on the "street side" of a lady when your on the sidewalk...things like that. This stuff is not taught anymore, that I know of....and it's a shame, really.

I'd add churches to that list as well; it may be a "public" building, but I remember the scoldings I got as a boy for not removing my cap when going into the local church!

And thank you, Scott, for the clarification!

I'm just now coming to this thread, and I'm really glad it's here. Please pass this along to every teenage boy you know, especially if he's in his twenties or thirties. "Hats" includes baseball caps.
My late uncle couldn't abide hats being worn indoors, and once upon a time his son to supper with a baseball hat on. It didn't remain on his head very long, and he didn't remain upright in his chair very long, either. Now my uncle's grandson doesn't even remember he's wearing a hat, much less make any attempt to remove at meal time.

Things change, but sometimes they can change back, and for the better.

Usage does differ depending on which side of the Atlantic you are on.
Wearing a hat in a lift (elevator) is frowned upon in Britain and South Africa (and, I imagine, in Australia and New Zealand).
The rules for saluting in uniform have changed in South Africa.
It used to be (following British tradition) that soldiers only saluted when hatted. If indoors and hatted, they would still salute – unlike sailors, who do not.
An acquaintance who served in the South African Navy found it very strange when military policement were sent to Simonstown to pick up a sailor and take him to detention barracks in Wynberg (Cape Town), they kept their caps on when entering the dockyard police office, and saluted.
The army salute was also the traditional British one: longest way up, shortest way down. When at the salute, the hand and elbow were both fully extended.
But in the new South Africa, our troops use the US salute, which to us old-timers is terribly sloppy.
I believe the rule still stands in South African that a civilian (or a soldier in mufti) never salutes. I recall seeing the woman mayor of Grahamstown acknowledging a salute by tipping her bicorn hat.
Cygnus, you are only too right about hat-wearing in church. But I recall that a minister’s son went unreprimanded when he wore a cap in our church.
Regards,
Mike

Amen! Other lost courtesies include opening doors for ladies, helping a lady into and out of the car, helping a lady into and out of her coat, pulling the chair to seat a lady at the table, rising when a lady enters or leaves, etc. And there are even special rules of etiquette for those of us who have wives with very long hair.

I daresay that almost no one knows these courtesies any more, and will give you strange looks if you exercise them. But on the rare occasion that a woman recognizes the courtesy and expresses her appreciation, it's worth it.

Unfortunately, some extreme feminists decried this behavior as just another way to oppress women and men became a little "gun shy".

Personally, I will hold a door for ANYONE, but especially those who might need a little help (elderly, disabled, etc.). For me, it's just being polite, but it's also amazing how many people don't even consider it.