Brent crude was up $1.21, or 3.7%, at $33.71 a barrel by 0912 GMT, its highest level since mid-April.
U.S. West Texas Intermediate (WTI) crude was up $1.59 or 5.4% at $31.02 per barrel, its highest since mid-March.
"Optimism on the demand side of the oil equation has helped prices climb further, with gasoline demand coming back as governments ease confinement measures," said Rystad Energy’s senior oil markets analyst Paola Rodriguez Masiu.
Summer weather is enticing much of the world to emerge from coronavirus lockdowns. Shops and restaurants prepared to reopen in Italy on Monday, while other centres of the outbreak such as New York and Spain gradually lifted restrictions.
The positive sentiment in the market was reinforced as U.S. Federal Reserve Chairman Jerome Powell issued an optimistic outlook for economic recovery later this year.
"Assuming there is not a second wave of the coronavirus, I think you will see the economy recover steadily through the second half of this year," Powell said Sunday night in broadcast remarks.
The June WTI contract expires on Tuesday, but there was little indication of it repeating a historic plunge below zero last month on the eve of the May contract's expiry.
However, analysts cautioned that demand was not expected to recover to pre-coronavirus levels any time soon.
"Clearly the fundamentals in the market are improving, but we continue to believe that the market is rallying too much too soon, with the risk that further strength will only prolong the supply and demand imbalance," ING analyst Warren Patterson said.
Also supporting oil prices are production cuts by the Organization of the Petroleum Exporting Countries (OPEC) and its allies, including Russia, a grouping known as OPEC+.
The world's top exporter Saudi Arabia announced last week that it would cut an additional 1 million barrels per day in June, while OPEC+ wants to maintain existing oil cuts beyond June when the group meets next.
Kuwait and Saudi Arabia have agreed to halt oil production from the joint Al-Khafji field for one month, starting from June 1, Kuwait's Al Rai newspaper reported on Saturday.
Production is also falling as U.S. energy firms cut the number of oil and natural gas rigs operating.