That's totally up to you.
As the DM, it is entirely up to you to decide if the circumsances merit advantage on a check:
The DM can also decide that circumstances influence a roll in one direction or the other and grant advantage or impose disadvantage as a result.
Here's how I do it.
You describe two scenarios, one where the PCs try to influence the roll before the check, and one where they try to get more intimidating after they have failed the check. I have encountered both with my players.
In your first scenario:
Let's say that the PC casts disguise self, to grow (slightly) larger, growing horns and flaming eyes, and look like a demon/devil. Maybe use thaumaturgy to make his voice powerful, and shake the ground.
This is absolutely the kind of thing I give advantage for; my players have often used both of these tactics.
Your scenario is a little bit different: the PCs have already failed the check once, and want to try again with a little extra from disguise self or thaumaturgy. I do not call for another intimidation check without something extra. What I mean is that once you have failed the check, you are not getting another try unless you change your approach by using something like thaumaturgy to make yourself more intimidating. And when you do it, I do not give advantage, and the DC is usually higher for the second check.