Beshear on Monday morning confirmed 30 deaths, followed by five more in an afternoon briefing, when he said there would be yet more to come. Hours later he confirmed on Twitter there had been two more deaths.
Authorities continued to work to rescue residents and provide food and shelter for thousands who had been displaced. Efforts have been hampered by weather conditions, officials say.
Many residents had been unprepared for heavy downfall overnight, leading to more deaths, Beshear said. For people remaining in Eastern Kentucky, he advised seeking higher ground ahead of evening storms.
"It is a continuing natural disaster. We are still searching for people," Beshear said in a CNN interview. "A large amount of grief throughout Kentucky."
The National Weather Service forecasted several rounds of continuing showers and storms through Tuesday.
Beshear, who declared a state emergency last week, said over the weekend that authorities would likely "be finding bodies for weeks" as teams fanned out to more remote areas.
Days of heavy rainfall - described by Beshear as some of the worst in the state's history - caused some homes in the hardest-hit areas to be swept away. Video clips posted online showed rescue teams guiding motor boats through residential and commercial areas searching for victims.
The Wolfe County Search and Rescue Team on Sunday published footage on Facebook of a helicopter lifting an 83-year-old woman from the roof of a home almost completely submerged. This was part of a five-person rescue.
At least 16 deaths were reported in Knott County alone. The bodies of four children, between ages 18 months and eight years, were recovered Friday afternoon. A fast current had swept them out of their parent's grip, a family member told the Lexington Herald Leader.