Princes of the Apocalypse would be the best fit. Period. It should have been marketed as the follow-up to Phandelver.
It's written by the same author, so you know what you're getting.
The overland maps for the two adventures overlap. Very easy to "stumble upon" the new adventure or to receive the message that help is needed. Also, since the official hook (which can be easily ignored) is that a dwarven envoy has gone missing, the fact that the PCs have already proven themselves as allies to at least one dwarf works well.
PotA has a few low-level side treks designed just to bring the PCs up to 3rd-5th level. These can be safely ignored, if your party is already at that level, without losing anything of consequence. On the other hand, the fast advancement to 3rd level in 5E means that you can safely use these without leveling up your group too much, if your group is big on exploration, role-play, or you just want them invested in the area.
My group went from Phandelver to PoA without any fuss, whatsoever. Since we're using Eberron, and I set the adventures in different geographic areas, that's a statement purely about the mechanical synergy of the two adventures. Using the flavor that comes from the Realms would only improve integration.
Tyranny of Dragons can be rejiggered to work with a 5th level group, but you have to skip the initial hook and miss out on some of the escalating story. I, personally, think that ToD is an extremely weak adventure and wouldn't recommend it even if you were creating characters fresh. YMMV on that, though.
I haven't read through Out of the Abyss, though it sounds like a decent adventure. The big red flag of which I'm aware is that it has an assumed start of 1st level/pre-adventurer PCs being captured and starting in a cell. That would be a huge bummer for anyone who'd gathered 5th level worth of stuff, and it could really unbalance that part of the adventure to do otherwise. My understanding is that the first half (or so) of the adventure is the captured PCs making their way free, so it can't be easily skipped.