In the case of Hex

PHB page 251

You place a curse on a creature that you can see within range. Until the spell ends, you deal an extra 1d6 necrotic damage to the target whenever you hit it with an attack. Also, choose one ability when you cast the spell. The target has disadvantage on ability checks made with the chosen ability.

There are three types of rolls in the game: Ability checks, saving throws and attack rolls. You seem to think that the Hex disadvantage applies to saving throws which is not the case. There will be no disadvantage on saving throws due to Hex on subsequent spell effects.

When does concentration end?

If we ignore the oversight on the Hex spell, your question asks about when exactly concentration is lost, or rather: does the second spell take effect before the first one ends. Refer to the following text from rules on concentration:

PHB page 203

You lose concentration on a spell if you cast another spell that requires concentration. You can't concentrate on two spells at once.

The last sentence makes it very clear that as soon as you cast the second concentration spell, the first one ends. The rules do not allow you to have two concentration spells in effect at the same time. Thus, the effect of the first spell cannot affect the second spell. In other words: The first spell ends before the second spell takes effect!

Yes, a warlock can keep the hex spell up outside of combat.

In-game, a turn is defined to have a duration of 6 seconds. When not in combat, you are not in initiative order, so turns are not structured rigorously. Nevertheless, your "turn" out of combat occurs whenever you have the opportunity to do something. As long as you have 6 seconds to spare, you can use your bonus action, and therefore you can redirect your hex.

As you noted, this is also implied by the extremely long durations that Hex can have. Nobody will ever be in a combat that lasts 8 hours (that's 4,800 turns!), so the only reason for the duration to be that long is to allow it to be maintained across multiple combats (or applicable social encounters) throughout an extended period. This is also mitigated by the Warlock's spell economy: they have very few slots to spend, so the requirement to spend a slot on Hex mitigates what could be perceived as an overpowering effect.