The songs themselves aren’t really sequels, but the Smashing Pumpkins’ Perfect was considered sort of a spiritual successor to 1979, so they made the video into a sequel, bringing most of the kids back a couple of years later.
From the Motown hit factory.
Mary Wells, My Guy followed immediately by the Temptations, My Girl.
The unfortunate answer to Barry McGuire’s Eve of Destruction, The Spokesmen and Dawn of Correction.
The Four Tops did exactly the same tune twice. The second one is called “The Same Old Song,” but for some reason the first one is not coming to mind.
Now I remember–“Sugar Pie, Honey Bunch.”
I’m looking for the cite now. IRRC, Sting has said in at least one interview that he wrote IYLSSTF as an antidote for EBYT. So far, I’m only finding websites that are referring to the interview, but not a cite for the interview itself. It’s possible I fell pray to an urban legend.
Glass Onion was a sort-of sequel to Strawberry Fields:
It also refers to Lady Madonna, I Am The Walrus, The Fool On The Hill, and Fixing A Hole (Where the Rain Gets In), but not in a sequelly way I guess.
Dream Theater’s concept album Metropolis Pt 2 is a sequel to the song (surprisingly) “Metropolis Pt 1”. So not quite a song but along those lines.
Thank you for mentioning EPMD. I haven’t thought of them in a long time. Now I am gonna YouTube them and get some of that good old school in my system.
As to the OP, the “Roxanne, Roxanne” hip-hop series is famous to hip-hop fans. It was started by U.T.F.O., but my favorite one is the one by Roxanne Shante.
Britney Spears. 
Her first hit, “Baby One More Time,” contains the lyric, “My loneliness is killing me.”
“Stronger,” a song from her second album, contains the lyric “My loneliness ain’t killin’ me no more, I’m stronger.”
Wilbert Harrison had a song “Goodbye Kansas City” a while after his version of “Kansas City” became popular.
This is more a “response” than a sequel, but bears mentioning, Koko Taylor’s “I’m a Woman” to Bo Diddley’s “I’m a Man.”
Hands Open by Snow Patrol references - Sufjan Stevens “Chicago”
On a more obscure note, the Hoodoo Gurus did two songs “Hayride to Hell” and “Hayride to Hell - Part II”.
The latter song changes all of the assumptions you would make after hearing the first song.
Neil Young’s “Southern Man” got Lynyrd Skynyrd mad, leading to, “Sweet Home Alabama”.
Not sure about the spelling there. But y’all know who I mean, I ween.
As did the OP, who mentioned exactly those songs in the very first line of his post.
Oops. Not sure how I missed that, but evidently I did. I’ll just crawl back under this here rock. :o
The mother of all song sequels: The Roxanne Wars
Death Cab for Cutie’s “Company Call Epilogue” follows…“Company Calls.”
The Moody Blues -
- In Your Wildes’t Dreams
- I Know You’re Out There Somewhere
John Fogerty’s I Saw It On TV is a sequel to his earlier Who’ll Stop The Rain (from his CCR days). And Deja Vu (All Over Again) could be considered another sequel, or the third in a trilogy.