The 19-year-old was walking in Larkhall, in the lane between Robert Smillie Crescent and Westerton Avenue, when he was approached by a group of three men around 3pm.
After what police said was a “brutal attack”, the trio made off towards Robert Smillie Crescent and the victim made his way home.
He did not visit hospital until the next day.
Officers are now making CCTV inquiries into the South Lanarkshire attack.
Police are also asking people to check dash-cam or home security footage for the group, who were wearing tracksuit bottoms and hoodies.
Detective Constable Stewart Walton said: “This was a brutal attack and the man, who did not realise how serious his head and facial injuries were until the following day, has since had to attend hospital on multiple occasions for treatment.
“Officers are carrying out inquiries and are still trying to establish a motive.
“We are appealing for anyone who may have seen this group of men, only described as wearing tracksuit bottoms and hoodies, in the area on New Year’s Day.
“You may also have dash-cam footage or personal house security camera footage that could help us, please think back.
“Violence will not be tolerated in our communities and we will do everything we can to trace those responsible.”
Anyone with information should contact Cambuslang CID on 101 or Crimestoppers on 0800 555 111.