Jones went under the knife last July, and the typical recovery time from Tommy John surgery is 10-14 months.
"I'm trying not to get too far ahead of myself, but I've been throwing off the mound without pain, and that's a huge accomplishment from last year," said Jones, who added he began throwing breaking balls off the mound earlier this week.
Jones pitched in only two games last season while also dealing with a back injury that is now healed, he said.
As any pitcher who has undergone Tommy John surgery can attest, the long, grueling rehab is more of a mental challenge than a physical one. Jones has been diligent in not trying to do too much, crediting the front office, coaching staff and training staff for reminding him the journey is a long one.
"Just have to take my time and get through each day healthy, that's the main thing," Jones said. "Do my throwing program or bullpen for that day and check it off."
Jones is happy with the process thus far, and hopes the rest of it continues to go as smoothly.
"I'm coming along pretty strongly, and hopefully the rest of the rehab goes just as well," Jones said.
Cash Kruth is a reporter for MLB.com. Follow him on Twitter at @cashkruth. This story was not subject to the approval of Major League Baseball or its clubs.