He believes the oil market recently rallied, in part, because of seasonal jitters over supplies, strong demand and the response at the pump.
"It was such a steep rally, such a steep uptrend on the chart … that once we break down below it to about even $60 a barrel, we can go a long way very rapidly and get back down at least to the low-$50s," he added.
U.S. crude futures settled up 53 cents, or 0.88 percent, at $60.93 a barrel on Wednesday. It rallied more than $2 to the year's high of $62.58 a barrel earlier. Brent futures were last unchanged at $67.52 a barrel, having reached a 2015 peak of $69.93 earlier.
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The spike came after the U.S. Energy Information Administration said crude stockpiles fell by 3.88 million barrels last week.
However, noted investor Dennis Gartman said while that report appeared to be bullish, a closer look at things like gasoline supplies showed that "the actual aggregated total inventory was actually not as bullish as the number would have indicated."
Because of that, the editor and publisher of The Gartman Letter said he ended the day short "a little bit of crude."
He also thinks that U.S. oil companies are ready to bring wells online that have already been drilled.
"They're going to be brought back on stream very quickly. So I think production numbers are going to pick up rather dramatically," Gartman said.
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Kilduff agrees that U.S. oil companies are "chomping at the bit" to get back in, noting that EOG Resources has said that if oil hits $65 per barrel it will re-energize its drilling.
Kilduff, who in February said he believed oil was likely to head as low as $30, said all the elements that brought oil down to the low-$40s remain in place.
For example, U.S. production has not gone down "one iota," he noted.
"We're a million barrels over last year. If we get more Iranian barrels on the market from the easing of the sanctions, the perfect storm will re-emerge—particularly later in the summer when our refiners go into maintenance to gear up for the winter heating season," he said. "You could see a real glut develop once again and it will get ugly."
—Reuters contributed to this report.