The flying fish (Exocoetus) enters the LRT alongside the swordfish

Nesting flying fish (Exocoetus) with swordfish (Xiphias)
(Figs. 1–3) in the large reptile tree (LRT, 1542 taxa; subset Fig. 4) seems like an odd pairing. Even so, no tested taxon is closer to swordfish than flying fish…

…until you add the needlefish (Tylosurus),
(Fig. 5) and then it all seems to make more sense visually.

Exocoetus volitans (Linneaus 1758; up to 30m ) is the extant blue flyingfish, here related to the swordfish, Xiphias. Exocoetus travels in schools or schoals. Sometimes they exit the water to avoid predators. Juveniles have a relatively shorter torso. Hatchlings are slow-moving and tiny. Note the antorbital fenestra and large lacrimal, as in Xiphias. Distinctly flying fish have a jaw joint directly below the orbit. The coracoid is larger than the scapula, raising the pectorl fins.

Xiphias gladius (Linneaus 1758; Gregory and Conrad 1937; up to 4.5m in length) is the extant swordfish, derived from the barracuda, Sphyraena. 1cm long hatchlings more closely resembled little barracudas, then little sailfish before reducing the long dorsal fin. The sword is not used to spear, but to slice and maim smaller fish traveling in schools. The pelvic fins are absent. Larger females produce more eggs, up to 29 million.

Please compare the juvenile swordfish, with Lepisosteus, the long-nose gar, which nests as a sister and shares many traits, including long toothy jaw and an elongate body.

Tylosurus acus (Lacépéde 1803) is the extant needlefish, a long-snouted, toothy sister to the flying fish, Exocoetus, in the LRT and other cladograms. Note the divided naris and new bone identities compared to the Gregory 1938 drawing. Distinct from related taxa, the pectoral fins are set further posteriorly on the torso. This speedy taxon somewhat bridges the gap between flying fish and swordfish. Compare this adult to the juvenile swordfish (above) and to the ancestral barracudas and pikes, Sphyraena, and Esox.

Readers may recall
an unrelated, though convergent, Late Triassic flying fish, Thoracopterus, which earlier nested with extinct Xiphactinus and extant anchovies, like Engraulis, in the LRT.

Iniopteryx, an odd chimaera/ratfish from the Late Carboniferous, also had large elevated pectoral fins, but those were made for display, not for gliding. That fish entered the LRT here.

Traditional fish classification
considers swordfish, needlefish and flying fish members of the broad clade Percomorpha (which one might imagine are perches and their descendants, but actually include cusk eels and their descendants. Otherwise traditional cladograms do not lump flying fish and swordfish together between pikes and garfish, as recovered in the LRT (subset Fig. 4).

If I missed a citation that predates this one
that supports this hypothesis of interrelationships, please send me the citation. It does not appear to be matched by genomic (gene/molecule) studies.

References
Gregory WK and Conrad GM 1937. The comparative anatomy of the swordfish (Xiphias) and the sailfish (Istiophorus). The American Museum Novitates, 952:1-25.
de Lacepéde BG 1803. Histoire naturelle des poissons. Tome Cinquieme. 5(1-21):1-803 + index.
Linnaeus C 1758. Systema naturæ per regna tria naturæ, secundum classes, ordines, genera, species, cum characteribus, differentiis, synonymis, locis. Tomus I. Editio decima, reformata.

wiki/Xiphias
wiki/Exocoetus_volitans
wiki/Tylosurus