Joe Haldeman'sForever Peace is, as the name implies, a Spiritual Successor to The Forever War despite taking place in a very different setting and, indeed, having very different basic assumptions about the setting. It reads as a more "mature" attempt to understand war by probing questions about the inevitable results of technological advances in warfare in the future that The Forever War glossed over so that its sci-fi war could be a clearer parallel to Vietnam.
Lewis Carroll's epic nonsense poem "The Hunting of the Snark" is a Spiritual Successor to the Alice stories, and includes a number of references to "Jabberwocky."
The Divine Invasion by Philip K Dick is a Spiritual Successor to his earlier novel VALIS: Valis appears in both books, the fictional film "Valis" exists in both, and they have similar Gnostic themes, but The Divine Invasion is not, strictly speaking, a sequel. A third novel The Owl in Daylight was going to be written by PKD as another Spiritual Successor to round out the "Valis trilogy", but he died before writing it.
The relationship between VALIS and its earlier version Radio Free Albemuth is actually a much more typical example of the trope, as they heavily overlap in themes but are emphatically not part of the same Verse. Or they would have been if PKD hadn't left Radio Free Albemuth unpublished during his lifetime, so that it came out about five years after VALIS.
In the extras to the DVD of Dreamcatcher, Stephen King notes that the book (and subsequent film) can be seen as a Spiritual Successor to The Body/Stand By Me.
Lois Lowry's book Gathering Blue is a "companion novel" to The Giver; it's another postapocalyptic novel which may be in the same universe, but shows a society that has gone the opposite direction.
Messenger is similar. It has a number of the same characters (Mattie, Kira, Jonas, Kira's father, etc) and Jonas at one point alludes to his previous village and how the people there made peace with him after he left, but most of the plot focuses on the corruption of the new society that Jonas has built.
Isaac Asimov's Fantastic Voyage II: Destination Brain is a... complicated example. The name indicates that it is an actual sequel (which would disqualify it), but as it turns out it is essentially a remake: taking the basic concept of Fantastic Voyage (miniaturization technology as a potentially crucial part in the Cold War and an attempt to use it to save the life or knowledge of someone who has made a critical breakthrough but failed to communicate it before falling into a coma), and then writing his own story around it, free of the constraints he was acting under when he wrote the novelization to the movie and able to update the science to 1980s standards.
Glamorama is a Spiritual Successor in many ways to American Psycho. While Bret Easton Ellis has been accused of writing about the same subjects(shallow, drug addled rich people) over and over again, and frequently using over the top violence to satirize mindless consumerism, Glamorama has a very similar surreal style comparable to American Psycho that his other books don't have since they're more grounded in reality.
Warrior Cats has many similarities to The Book of the Named - so much that people were claiming that the Ratha series copied Warriors, until it was pointed out to them that Ratha's Creature was written in the 1980s and Into the Wild came out in 2003.
Nineteen Eighty-Four is arguably this to Sinclair Lewis' It Can't Happen Here, published fifteen years earlier. Both novels depict a formerly democratic western nation that succumbed to totalitarianism. The protagonists of both novels chafe under totalitarian rule and rebel through the written word. Both men find solace in secret romantic relationships with women who are both their soulmates and co-conspirators; Winston falls in love with Julia while Doremus has a secret affair with Lorinda. Finally, both protagonists find themselves incarcerated and tortured for their rebellion against the state.
Heartlight by T.A. Barron is so similar in both style and themes to A Wrinkle in Time that Madeleine L'Engle herself has given the novel praise.
Jack London's White Fang is the spiritual successor to his The Call of the Wild. Both are xenofictional stories about dogs on the edge of civilization, one about a wild dog being tamed and the other about a domestic dog going feral. They're generally even published together in a single volume, as if the former were an actual sequel.
Harry Potter is definitely this to The Chronicles of Narnia; despite having very different plots, both are series that heavily utilize magic, follow strong children who save their world, and provide an allegory that is well-written enough to appeal not only to its young target audience, but to spark the interest of adult readers as well.
The Power of Five is this to Anthony Horowitz's unfinished Pentagram series from the 1980s.
Killing Pablo is a rare nonfiction example. Written by Mark Bowden and featuring the exploits of General Bill Garrison, it reads much as a sequel to Black Hawk Down.
John Varley's The Golden Globe is a combination homage and spiritual successor to Robert A. Heinlein's Double Star. The protagonist in both is a highly skilled and intelligent but down on his luck actor who used to be famous, and now lives partly on the wrong side of the law while still being obsessed with his craft (something drilled into him by his father). The characterizations and habits are essentially the same, and they also deliberately share a similar first person narrative style, from the perspective of the character writing out his experiences after the fact.
The Rainbow Magic series is receiving one in the Magical Animal Friends series, written by the same author. This series revolves around Jess and Lily, who enter the mysterious Friendship Forest and rescue animals from the wicked witch Grizelda and her Boggit servants.
A case can be made that Lew Wallace's popular and acclaimed novel Ben-Hur serves as this to Alexandre Dumas' The Count of Monte Cristo. Wallace had cited it as one of his favorite stories and as an influence on his own work. It is visible in the parallels between the stories of Edmond Dantes and Judah Ben-Hur. Both are good well to do men who are eventually betrayed and wrongfully have their lives stripped away from them and are imprisoned in one way or another. Both however manage to eventually "rise from the ashes" so to speak an attain their freedom and go on a mission for justice/revenge.