Smith is singing with an extended portamento style. If you want to be precise, you can use a slur symbol between notes (but be warned, a slur between more than two notes will be read as legato or phrasing). He also slides down at the end of some phrases, which you can probably notate as a downward slur to a grace note on the end pitch.
However, the vocals on this piece are incredibly complex and will probably take you a lot of time, and you will likely never be able to fully re-create the vocals with another singer even with accurate notation due to the microtonality inherent in his style.
What is the end result you want? Ask yourself if you want to pay homage to Smith's vocals, or if you want to take them as inspiration only. Most singers spend years developing their own unique styles, and might find a score more accessible if you give them a more simple framework which they can add their own flair to.
If it's the former, I agree with @phoog: get the singer to listen to the track and learn the intricacies by rote (singers are trained in being able to pitch-match and will be able to follow the microtonal shifts more accurately than you can probably notate).
If it's the latter, I'd use his melody as well as the harmonic information in the instrumentals as a guide and write out a simple melody, possibly using stylistic text notation like "portamento", "ad libitum", "freely", etc., at the beginning.