The man, 29, was re-arrested on Thursday in connection to the baby boy’s death in hospital in December.
He was also arrested for section 18 wounding, along with a 28-year-old woman.
Temporary Detective Chief Inspector Mark Oughton, leading the investigation, said: “This is a tragic and extremely complex case. My team of officers has been working hard over the last few months to piece together everything we know about what happened in the lead up to the devastating loss of this baby.
“I know this incident had a significant impact on the local community and our thoughts continue to be with those affected.
“I would ask that people are mindful of what they post on social media, particularly in relation to any speculative comments, and bear in mind the impact this could have on the baby’s family during this incredibly difficult and traumatic time.”
Emergency services had been called to a house in Great Houghton on November 30, after reports of a baby going into cardiac arrest.
Enquiries into the death have been ongoing for a number of months, with the 29-year-old man first arrested on December 5 and released.
Both suspects have been bailed pending further enquiries following Thursday’s arrests.
Anyone with concerns or information should call 101 quoting incident number 851 of November 30, 2018.