"Two sources tell News 4 that [Jeffrey] Epstein may have tried to hang himself, while a third source cautioned that the injuries were not serious and..."

"... questioned if Epstein might be using it as a way to get a transfer. A fourth source said an assault has not been ruled out, and that another inmate was questioned. The inmate who investigators have talked to in Lower Manhattan facility has been identified as Nicholas Tartaglione, according to two sources. Tartaglione is a former police officer in Westchester County who was arrested in December 2016 and accused of killing four men.... The attorney for Tartaglione denied all the claims that his client attacked the financier, saying his client and Epstein get along well.... 'They are in the same unit and doing well,' said Bruce Barket, an attorney for Tartaglione. He said any claim that Tartaglione might have assaulted Epstein 'is absolutely not true.' Barket said Tartaglione and Epstein have been complaining about conditions inside the MCC including flooding, rodents and bad food."

From "Jeffrey Epstein Found Injured in NYC Jail Cell After Possible Suicide Attempt: Sources/Epstein was found semi-conscious with marks on his neck, sources said and investigators are trying to piece together exactly what happened" (NBC New York).

Add it all together and the most plausible theory seems to be that Epstein, wanting a transfer to a nicer jail (or release on bail), got his new friend Tartaglione to strangle him enough to make it look like a suicide attempt.