Some Solomon references/allusions (compiled by Sarah L. Schwarz):

On the Origin of the World 2.106, 19-107, 17 cites "The Book of Solomon" as a source for information on the demons and their effects: (find "Solomon" in page)

Testimony of Truth 70, 6-7 does not cite a book but references a story of Solomon imprisoning the demons in waterpots, etc.: (find "Solomon" in page)

Dialogue of Timothy and Aquila cites information not recorded in Kings but in Solomon's testament:

Babylonian Talmud (Gittin 68b) has the Solomon/Ashmedai cycle

Josephus' Antiquities 8.2.5, mentions Solomon's many writings and power over demons, and describes Eleazar's use of a Solomonic exorcism

references to Hezekiah's suppression of "book of the recipes" or "remedies":
[we should talk about this one!]

(less useful)

PGM IV 3009 adjuration of demons "I adjure thee by the seal which Solomon laid upon the tongue of Jeremiah and he spoke"

Origen, in commentary on Matt 26:63, says "it is customary to adjure demons with adjurations written by Solomon..."

Apocalypse of Adam 5.78, 27-79 cites a tradition of Solomon marshalling an army of demons:

Many bowl texts and amulets mention Solomon and may or may not reference these traditions. E.g., Supplementum Magicum 24, Schiffman & Swartz

Islamic traditions know Solomon and the djinn building the temple, and also Solomon's magic ring

On the Book of Remedies, attributed to Solomon:

These are some references to the Book of the Remedies (BR), purportedly a work
by Solomon which contained cures to all diseases, and was suppressed by
Hezekiah to prevent people from trusting in it rather than in God. References
are compiled from McCown's introduction (pp. 96-100) and Halperin's article
"The Book of Remedies, the Canonization of the Solomonic Writings, and the
Riddle of Pseudo-Eusebius" JQR 72 no 4 (1982) 269-92.

B. Ber. 10b:
"Our masters taught: King Hezekiah did six things: three [the Sages]
confirmed, and three they did not confirm.
Three they confirmed: he concealed the Book of the Remedies, and they
confirmed this; he destroyed the bronze serpent, and they confirmed this; he
dragged the bones of his father on a bed of ropes and they confirmed this.
Three they did not confirm: he stopped up the waters of the Gihon, and they
did not confirm this; he cut off the Temple doors and sent them to the king of
Assyria, and they did not confirm this; he intercalated Nisan in Nisan, and
they did not confirm this."

All these deeds of Hezekiah can be attributed to (canonical) scripture,
basically, accept the BR. By the middle ages, Jewish interpreters are refuting
the allegation that it refered to a book of Solomon's cures, so the idea was
out there, but there's a big gap in Jewish sources.

However, it does pick up in Greek Christian sources. Hippolytus, in his
commentary on the Song of Songs, references this: (find
Hezekiah in page). And it gets conflated and confused in Christian sources
throghout the middle ages.