"Whenever he leaves the house, he looks happy. He looks alert and often smiles," a source tells PEOPLE

By Aili Nahas and Aurelie Corinthios
Updated January 29, 2016 02:30 PM
Gregg DeGuire/WireImage ; Jason LaVeris/FilmMagic

brightcove.createExperiences(); Lamar Odom is making headway on his road to recovery after a near-fatal overdose.

The former NBA star, who was hospitalized Oct. 13 after being found unconscious in a Nevada brothel, was released from Los Angeles’ Cedars-Sinai Medical Center earlier this month and now stays in a leased house in the same gated community as his ex Khloé Kardashian, a source tells PEOPLE.

“Khloé continues to look after him,” the source says. “Khloé visits him at his house and Lamar hangs out at hers.”

The source says Odom, 36, receives therapy both at home and at a facility.

“He still needs help walking,” the source adds. “Lamar is not driving himself and instead has a driver. Whenever he leaves the house, he looks happy. He looks alert and often smiles. He is definitely looking more like the old Lamar.”

A Kardashian insider adds that while he “isn’t ‘back to normal’ by any means, he’s made so much progress since the overdose.”

“He can move around alone, and he can communicate much better. In the beginning, he couldn’t talk at all,” the insider tells PEOPLE. “He knows how much Khloé has been there for him, and he is so grateful. The therapy can be incredibly frustrating and he has hard moments, but Lamar is a former athlete. He’s strong. And he really wants to get better.”

Most recently, Kardashian, 31, gave an update on her ex’s condition when she dropped in on sister Kourtney‘s appearance on The Ellen DeGeneres Show on Monday via FaceTime.

“His long-term memory is great,” Kardashian told host Ellen DeGeneres. “His short-term memory isn’t so good, but I mean, if I met someone earlier today I probably wouldn’t remember either, so that’s understandable. But he’s great. He’s hung out with Kourt’s and Kim‘s kids. He loves family time.”

“I think it makes him feel good to be around everyone too and to just be in a happy environment over obviously where he was,” she added.

Kardashian also revealed that they’ve been explaining the situation to him: “Every day he’ll ask me: ‘So what happened to me?’ ” Kardashian said.

“Like today specifically, he was like, ‘How many tubes did I have in me again?’ And I’m like, ‘I don’t know what’s too serious to tell you,’ but I’m honest,” she added. “But it’s good that he wants to know these answers. Before he didn’t want to know, he wasn’t ready to know. So it’s great that he’s at the point that he’s ready to know everything happened.”