And in that moment all the trumpets were blown, and the King Elessar went forth and came to the barrier, and Húrin of the Keys thrust it back; and amid the music of harp and of viol and of flute and the singing of clear voices the King passed through the flower-laden streets, and came to the Citadel, and entered in; and the banner of the Tree and the Stars was unfurled upon the topmost tower, and the reign of King Elessar began, of which many songs have told.

Did Tolkien write anything concerning Middle-earth taking place after The Return of the King?
I find the matters leading up to his coronation to be more song-worthy than the little I know of his actual reign.

It later appears that there had been a 'hallow' on Mindolluin, only approachable by the King, where he had anciently offered thanks and praise on behalf of his people; but it had been forgotten. It was re-entered by Aragorn, and there he found a sapling of the White Tree, and replanted it in the Court of the Fountain. It is to be presumed that with the reemergence of the lineal priest kings (of whom Lúthien the Blessed Elf-maiden was a foremother) the worship of God would be renewed, and His Name (or title) be again more often heard. But there would be no temple of the True God while Númenórean influence lasted.

Also to be Prince of Ithilien, the greatest noble after Dol Amroth in the revived Númenórean state of Gondor, soon to be of imperial power and prestige, was not a 'market-garden job' as you term it. Until much had been done by the restored King, the P. of Ithilien would be the resident march-warden of Gondor, in its main eastward outpost -- and also would have many duties in rehabilitating the lost territory, and clearing it of outlaws and orc-remnants, not to speak of the dreadful vale of Minas Ithil (Morgul). I did not, naturally, go into details about the way in which Aragorn, as King of Gondor, would govern the realm. But it was made clear that there was much fighting, and in the earlier years of A.'s reign expeditions against enemies in the East. The chief commanders, under the King, would be Faramir and Imrahil; and one of these would normally remain a military commander at home in the King's absence. A Númenórean King was monarch, with the power of unquestioned decision in debate; but he governed the realm with the frame of ancient law, of which he was administrator (and interpreter) but not the maker. In all debatable matters of importance domestic, or external, however, even Denethor had a Council, and at least listened to what the Lords of the Fiefs and the Captains of the Forces had to say. Aragorn re-established the Great Council of Gondor, and in that Faramir, who remained by inheritance the Steward (or representative of the King during his absence abroad, or sickness, or between his death and the accession of his heir) would [be] the chief counsellor.