If I am right, "white spirit" is an odorless, mostly paraffinic, hydrocarbon solvent with a boiling range of 153 to 197
oC. Not anymore if it contains mercaptans. Of those in this boiling range one could mention n-amyl mercaptan (BP: 126
oC) and heavier. In general, these are objectionable because they impart odour and may cause darkening of cooked varnishes and resins. Specifications for spirits ask for mercaptans to be NIL, or 2 ppm as a maximum, thus your intention of just diminishing their concentration must be very "site-specific".
Depending on their use (architectural finishes, painting, etc.) mineral "spirits" are classified as: regular, odorless, high-flash, and high-solvency.
The solvent properties that interest you were not given. If you are just looking for similar boiling points, "regular" (Stoddard) solvent, and Shellsol D40 may be acceptable additions. In this sense, a "high-flash" aromatic naphtha with an IBP 154
oC and a dry point of 177
oC devoided of mercaptans, may be applicable. If your "spirit" follows a given specification you should look for more details than just a distillation range. Good luck.