Readers were first introduced to Andrew Wiggin in 1985. Many science fiction readers know him better by his nickname Ender. Orson Scott Card released the novel Ender's Game in 1985. It went on to win the Nebula Award as well as the Hugo Award for the best Science Fiction novel of the year.
The chronological sequel, Speaker for the Dead, written just a year later, went on to win both awards as well. This story takes place after the events of Ender's Game. It is a sequel by definition.
The more interesting "sequel" to me however is the one written to take place during the exact same time frame as Card's original. The book Ender's Shadow was written in 1999 as a companion novel to Ender Wiggin's first adventure.
Writing a companion novel is usually the last technique used by writers, or movie directors for that matter, to continue the story of a beloved character/world. First comes a sequel. This is obvious. Continuing where we, the reader, left off previously and telling us what happens next is the common thing to do. If readers loved a character or enjoyed hearing about a tale set in a distant land, they will want to hear how things progressed.
If a sequel is not an option for some reason, say the main character dies, then the next best choice is writing a prequel. Again, obvious. Tell us how this character came to be who we saw them as. Tell us the back story and past events that shaped our main character's life.
Of course sequels and prequels are child's play. The writing of a companion novel is rarely seen because it is hard to make it interesting. Imagine reading a story you liked. Now imagine the author trying to write another novel that takes place during the exact same time frame, with the exact same characters and covers the exact same events. Would this be at all interesting to read? Simply writing from a new point of view is nice but it does not totally solve the problem. Just look at the movie Vantage Point for an idea of how that can fail.