Here’s a descent list of song that are in scripture that is not within the Psalter:
(2) The Song of Moses and Miriam (Exod. 15)
Spring up, O well! (Num. 21:17-18)
The Mosaic Song of Witness (Deut. 31:19-32:44)
The Song of Deborah and Barak (Judges 5)
The Book of Jashar (2 Sa 1:18)
Songs of Solomon (1 Kings 4:32)
The statutes of the Law were sung (Ps. 119:54)
The Song of the Vineyard (Isa. 5:1-7)
An Eschatological Song (Isa. 26-27)
The Prayer of Habakkuk “on shigionoth” (Hab. 3)
Mary’s Magnificat (Luke 1:46-55)
The Song of Zechariah (Luke 1:67-79)
The Angelic Doxology (Luke 2:14)
Simeon’s Nunc Dimittis (Luke 2:28-32)
A New Song (Rev. 5:9-10; 14:3)
The Song of Moses and of the Lamb (Rev. 15:3-4)
The Hallelujah chorus (Rev. 19:5-7)
Charismatic hymnody (1 Cor. 14:15, 26)
(And we should not forget those possible hymn fragments in Scripture (Luke 1:46ff; Phi 2:6-11; Col 1:15-20;1 Ti 3:16; Eph 5:11; Rev 4:8, 11, 5:9-10.) as well as the record of early church history.)
It cannot be proven that these songs were actually sung corporately or if they were sung at all.
The word ‘song’ does not denote singing in every respect:
2172. זִמְרָה zimrah (274b); from 2167; melody, song (in praise of Yah):—melody(2), song(3), sound(1).
Robert L. Thomas, New American Standard Hebrew-Aramaic and Greek Dictionaries : Updated Edition (Anaheim: Foundation Publications, Inc., 1998).
Consider Isaiah 12:2
2 Behold, God is my salvation;
I will trust, and not be afraid:
For the Lord JEHOVAH is my strength and my song;
He also is become my salvation.
The Holy Bible: King James Version, Electronic Edition of the 1900 Authorized Version. (Bellingham, WA: Logos Research Systems, Inc., 2009), Is 12:2.