Keep in mind too that there was a major event in that intervening 60 years: The Necromancer was driven out of Dol Guldur by the White Council, whereupon he revealed himself to be Sauron and reclaimed Barad Dur. There’s enough material there to make for an exciting movie all by itself.
Stuff that happens in those “60 years” for the second movie:
2941 Bilbo goes on his adventure and finds the one ring
—————————————60 years-——————————
2941 Expulsion of the Necromancer(Sauron)from Dol Guldur by the White Council.
2942 Return of Sauron to Mordor.
2951 Rebuilding of Barad-dûr begins.
2951 The Nazgûl are sent to reclaim Dol Guldur.
2951 Aragorn is told who he really is by Elrond
2953 The last meeting of the White Council.
2956 Aragorn meets Gandalf and starts protecting shire
2957-2980 - Aragorn using the name of “Thorongil” serves in the armies of King Thengel of Rohan (Theodins father), and Steward Ecthelion II of Gondor (Denethors father)
2968 Birth of Frodo Baggins in the Shire.
2978 Birth of Boromir of Gondor.
2980 Frodos parents die in a boating accident.
2982 Birth of Meriadoc Brandybuck
2983 Birth of Faramir of Gondor
2983 Birth of Samwise Gamgee.
2989 Bilbo adopts Frodo
2989 Balin attempt to retake Moria.
2990 Birth of Peregrin Took.
2994 Balin is killed in Moria
2995 Birth of Éowyn in Rohan.
—————————————60 years———————————
3001 Bilbo Baggins’ Birthday Party and farewell feast.
Stuff technically outside the 60 years but missed in LOTR movies:
3007 Aragorn mother passes away
3014 Saruman begins to weaken Théoden
3017 Gollum is released from Mordor; Aragorn finally tracks him down in the Dead Marshes, and brings him as a captive to King Thranduil’s halls in Mirkwood (Legolos’s father)