Focusing on the musicality:
* Song-like: This directly emphasizes the melody and rhythm.
* Melodious: Emphasizes the musical quality.
* Choral: Suggests a singing or chanting quality.
* Rhapsodic: Implies a passionate, overflowing style.
Focusing on the content:
* Subjective: Highlights the personal feelings and thoughts of the speaker.
* Emotional: Stresses the expression of feelings.
* Intimate: Suggests a personal and heartfelt tone.
* Reflective: Emphasizes the contemplative nature of the poem.
Focusing on the form:
* Free verse: If the poem lacks a strict rhyme scheme or meter.
* Metered: If the poem uses a specific metrical pattern.
* Rhymed: If the poem uses rhyme.
* Sonnet: A specific form of lyric poetry (but not all lyrics are sonnets).
The best alternative for "lyrical poem" will depend on the specific qualities you want to emphasize.