Emphasized pitches in the voice part at the beginning trace a line from ^5 to ^#7 in D minor, though the underlying harmony wanders far afield:
For the remaining lines of the first verse, the line sinks back through C-nat5 to the piano's pianissimo B3 (bar 13). I have not marked it but also notice an ancillary line bringing G4 (bar 8) through F4 (bars 9-11 in the piano, bar 11 in the voice) to E4 in the voice (bar 12; then repeated by the piano in the following bars). This register persists as an inner voice during the second verse.
For the second verse, the voice puts D: ^6 in the fifth octave (bar 17), then on a somewhat tortured path eventually finds it way to the structural cadence and D5 (bar 27). Given the shifting movements of the harmony, even in this march section, I hesitate to ascribe any particular figure to the voice's background shape, other than it rises overall.
References: The most important source is Sarah Hibberd, ed., Melodramatic Voices: Understanding Music Drama (Farnham, UK; Burlington, VT: Ashgate, 2011). The scholar who has written most extensively about melodrama is Jacqueline Waeber. See her Musique dans le texte: Le mélodrame, de Rousseau à Schoenberg (Paris, 2005). Some examples of melodramas of the type I mention written by contemporaries of Wolf include Franz Liszt, Lenore, S.346 (1857–58); Carl Haslinger, Der Bettler vom Rialto, Op.124 (1868?); and Siegfried Ochs, Der Handschuh (1883).