The left-hander has struggled with injuries to his arm, knee and shoulder and has played just 12 matches in the past year.
Those injury problems began at Wimbledon last year, when he was having the best run of his career after reaching the 2024 US Open semi-finals and rising to a career-high fourth in the world rankings.
But, after making his comeback on the south coast, Draper said he was in a “very stable place” with his injuries, which included bone bruising in his serving arm.
“Knee tendonitis doesn’t go away overnight, but I’m in a really good position to be very explosive,” he told BBC Sport.
“I feel like I’ve got full power and full capacity back again which is a massive feeling for me after not having that for a very long time.”
And Draper is glad to have Murray in his corner after an “incredibly difficult” year.
“Andy, truthfully, is one of my biggest inspirations. When I was younger I remember him playing all the big events and being there to watch him,” Draper said.
“He believes in my game, he believes in me as a person. That’s what I really need right now coming back from the year I have had.”
Fellow Briton Pinnington Jones came from a set down to defeat Argentina’s Marco Trungelliti earlier on Monday.
In the women’s competition, Britain’s Francesca Jones was brushed aside 6-2 6-2 by former French Open champion Jelena Ostapenko.
Teenager Hannah Klugman suffered a narrow 7-5 5-7 7-5 defeat by Czech 19-year-old Tereza Valentova.
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