However, "in the November, December period, we're going to see a different dynamic set in as we have much more difficulty keeping up the supply," John Hofmeister, now founder and CEO of Citizens for Affordable Energy, said in an interview with CNBC's "Closing Bell."

That's because, as one producer told him, the real downturn in the U.S. is not being felt yet but will be soon.

U.S. crude futures closed $2.66, or 6.25 percent, higher at $45.22 per barrel Friday. Oil saw its biggest one-day bounce since 2009 on Thursday, with North Sea Brent and U.S. light crude rising more than 10 percent.

U.S. oil rigs rose by one from last week to 675, according to Baker Hughes data. They still remain down 900 from last year.

Hofmeister, who has been saying that oil could go to even $70 or $80 later this year, is worried that the current surplus isn't enough to protect from a rapid runup in oil prices, noting that capacity cannot be rebuilt as quickly as it can be taken apart.

Consumers should also enjoy the current prices at the gas pump, he added, because he said it isn't going to last.

"I do think the next couple of months will be pretty good for consumers and then we start the long trek back up."

Analyst Helima Croft also sees oil rising later in the year, but for now she thinks it will have a hard time moving and staying sustainably higher.

She called the next six to seven weeks a "difficult" time because refineries will be shutting down for maintenance.

And if that sends oil prices lower, that wouldn't necessarily be a bad thing, the chief commodities strategist at RBC Capital Markets told "Closing Bell."

"Down is better if you want a fundamental clearing out of the market. You have to clear out the supply," she noted "So if you want a long-term price recovery, lower is better and staying lower."

Croft thinks the recovery is coming once maintenance season ends. She anticipates an average price of $54 per barrel for the fourth quarter.

—CNBC's Linda Sittenfeld and Reuters contributed to this story.