In an emergency situation, you're not going to want to take the time to switch into Grit before trying to establish hate, and unless you've just used Spinning Slash (which you normally, shouldn't be using as off-tank), you're not going to be able to get off Power Slash quickly enough.
Instead, start off by using Provoke, as you're doing, then if the mob isn't within melee range, use Unmend. If the mob is vulnerable to stun, hit Low Blow as soon as it's in range, to buy yourself a few seconds, then start spamming Unleash, while weaving in any offensive oGCD abilities that are up. You should very quickly generate enough enmity that you can safely take the GCD to get Grit up and then start your Power Slash combo.
If this is happening to you often, or you're in a fight where it's likely the main tank will die, you may want to consider spending part of the fight in Grit, or using your enmity combo without Grit some of the time. The idea being, if you're already in second place on the enmity list, then when the main tank dies, the mob is going to go after you, anyway.
The procedure is very similar for PLD. Just swap Shield Lob for Unmend, Flash for Unleash, and the Rage of Halone combo for the Power Slash combo. Two thirds of PLD's DPS combo is shared with their enmity combo, which makes it slightly easier to quickly switch from DPS mode to Tank mode. PLD also has the advantage of being able to Cover one person for 12 seconds.
WAR is definitely in the best position to deal with emergency situations of this sort, since Defiance and Deliverance are both oGCD, meaning they don't lose any time switching stances. But otherwise you would do much the same thing.
Most of the information in this answer is no longer relevant as of Shadowbringers. Tank stances are all oGCD now, enmity combos are no longer a thing, and provoke now generates a fair amount of enmity beyond just placing you at the top of the enmity list like it used to.
If you need to quickly take over because the main tank died, immediately weave in your tank stance and then pop provoke. That should be enough, but if it isn't you can throw out your ranged attack since it has a fairly large enmity multiplier.
While the main tank is still alive, you can use provoke followed immediately by shirk on the main tank every 2 minutes. Just be careful to avoid turning the boss while you do this, and don't do it if there's a tank swap required in the next 30 seconds. This has the benefit of ensuring you stay at number two on the enmity list, and giving the main tank a 25% boost to their current enmity every 2 minutes.