Second book syndrome has several definitions. One refers to the writing process of the second book after the successful publication of the first book. The writer fears that they won’t be able to write as good a book as the first. Or, they fear that the first book was a fluke, and they really don’t know what they are doing. Some authors become so frozen with fear that they can’t move forward with their writing.
The other definition refers to the time after the second book has been written and published. The reading audience discovers the writer’s paranoia about his skills were right, and the second book fails to deliver what the first book did.
Margaret Mitchell was so terrified of failure after GONE WITH THE WIND she reportedly decided not to publish another novel. Robert James Waller who wrote the phenomenally successful bestseller, THE BRIDGES OF MADISON COUNTY, proved to be a one-hint wonder. All his other books have failed to even remotely achieve the success of this novel. So, yes, second book syndrome does exist.
I’ve discovered a new kind of second book syndrome in series. In the last month I’ve read two urban fantasies that were a second in the series, and both failed badly for the very same reason. Up to sixty pages at the beginning of the book were nothing but clean up between the plot ending of the first book, and the plot beginning of the second book.
Minor unresolved problems were answered, and characters discussed their relationships and careers that have been changed because of the first book's events.
I imagine all this was vastly important to the author and some readers, but it was a massive brick wall to a majority of readers.
A second book should start like any book. The reader should be immediately shoved directly into the book with an important plot goal and engaging characters and should be kept there for the remainder of the book.
If you think some things should be clarified or expanded, wait until a bit later and have the character explain to a friend why she no longer works for the police, or why she fears her friends may be targeted by her enemy.
Also, let the reader intuit some changes. If they read the first book, they can usually guess why things have changed, and if they didn’t read the first book, they won’t care as long as you give enough information to cover the current situation.
And, remember, this holds true for all the books in a series. Successful series writers like JK Rowling or Charlaine Harris never maunder about at the beginning of each Harry Potter or Stookie Stackhouse novel, and neither should you.