The fact that she seeks her beloved, him whom her soul loves,
but does not find him. is not referring to her actually looking for him, but rather expresses her longing for him. She is longing for him, but her longing does not bring him closer.1
The phrase by night is the translation of the Hebrew word ballelot, which is plural. The literal meaning is probably night after night.
Together with the repetition of I sought him,
it shows the intensity of her searching and longing for the young man.2 Her immediate searching is frustrated. She does not find him, which is not surprising because the time has not yet come for them to be together.
1 On my bed by night I sought him whom my soul loves; I sought him, but found him not.