Is there a published adventure that fits well in terms of level required, geographical location, and general theme?
Many of the published adventures have been deliberately designed so that you can move from the Starter Set into them.
With the Tyranny of Dragons story, you can skip the first few chapters and jump into
Hoard of the Dragon Queen at the Carnath Roadhouse, which is aimed at 5th level PCs and is located in the Mere of Dead Men, not far from Phandalin.
Princes of the Apocalypse takes place to the east of Phandalin and includes story hooks to get Starter Set characters into the Dessarin Valley. The main plot does start at level 3, but you can just let them have a slightly easy time at the start.
Out of the Abyss is a bit more difficult to transition into. The first part is all about escaping the Underdark, and goes from levels 1-7. The PCs start as prisoners of the drow. I think it would be difficult to convince your players that their 5th level Starter Set characters got automatically captured by drow and taken to a prison some distance from Phandalin with no chance of escape on the way. The second part of the adventure starts at 8th level and involves going back into the Underdark to fight the demon lords. You could ostensibly give your players a few levels' worth of adventuring in the Phandalin area before sending them to see King Bruenor in Gauntlgrym (just north of Phandalin) at level 8.
Curse of Strahd includes story hooks to get players from the Sword Coast (via Daggerford, which is some way south of Phandalin, but you could easily rework that intro) into Barovia. 5th level PCs would have an easy time at first, but there's nothing wrong with that really.
Storm King's Thunder has a levels 1-5 intro adventure, but the real meat starts at 5th level. There are three starting points, one of which is quite close to Phandalin: Triboar. You could just give them some reason to go there and then kick things off when they get into town.