Yes, you can reuse characters.
There is nothing particular or special to Lost Mines of Phandelver that requires handling differently from any other content. Characters are not implicitly one-and-done - you don't throw away the character when you finish the book. Just because many of the hardcover adventures are designed to start at L1, doesn't mean they must do so.
Some modules even specifically call out how to blend them. For example, Storm King's Thunder has a section in the appendices that covers what to do with characters coming out of Lost Mines of Phandelver. The short version is "skip chapter one and start with chapter two", but there are a few more specific suggestions of how to work it into a story. Other modules have different hooks, or leave you room to develop your own hooks.
Balancing Being Overleveled
If you're starting a hardcover adventure with characters above the recommended level, you'll need to adjust for that (or at least be aware of it). Early encounters will certainly need the opposition to be buffed up in some way, and even later encounters may need it (the characters are starting overleveled, so they may be overleveled at the end, too). That's where DM discretion comes in.
This is also an opportunity to take into account player preference. Some players like the edge-of-your-seat danger and risk of hard or deadly encounters, and the tactical thought that goes into swinging the odds in your favor to win. Other players prefer waltzing into a room and slaughtering enemies by the score, making each encounter a blood-splattered mess. It's up to you as the DM to gauge your players and adjust the encounters to suit.
One of the easier ways to control character level is to not award experience points, but to assign additional levels as the story demands it. This is known as milestone advancement. This can relieve some of the burden of a party being overleveled, because you simply don't have them level up from encounters that aren't really a challenge.