At the end of LMoP, your PCs should be level 5. From the book's perspective, this means you can drop them right into Chapter 2: Rumblings. Page 231 of the book has several possible adventure hooks, but frankly, anything that gets the characters to nearby Triboar will be sufficient for getting the story kicked off.
If you look at the map from LMoP, Triboar is literally the "next town over", so there should be any number of reasons to visit. Once they've dealt with the Giant attacks on Triboar, they get offered some number of quests that have them wandering this part of the Realms. Depending on what happened in Phandalin, there is a lot of opportunity to weave in the PCs existing motivations with the progressing story line.
The book is designed to "evolve" the story for the PCs as the whole thing progresses. If the PCs get blindly dragged along for too long, Harshnag and the Eye of the All-Father do a lot of exposition.
Note that SKT is a big adventure in a large sandbox. Most adventures have the caveat that you should read the entire book before running it. This book has so many threads that I would recommend going a step further.
- The DM's Guild has a great "Guide to SKT" that gives a good overview and also organizes data that is otherwise spread across the book.
- The book has several call outs to Sword Coast Adventurer's Guide. If you want to add flavor to individual locations, this will be useful.
- Prep regional locations as appropriate. Some googling will bring up lots of Realms history associated with many of the places listed in the adventure. You should be able to find maps and significant NPCs for these places.