* Weak U.S. dollar supports oil prices
* U.S. drilling activity continues to rise (Updates with comment, refreshes prices; changes dateline from SEOUL)
LONDON, June 26 (Reuters) - Oil rose for a third straight session on Monday, as speculators took advantage of last week's drop to seven-month lows, although a relentless increase in U.S. supply and little evidence of a widespread drop in global inventories capped gains.
Investors in U.S. crude futures and options have increased their bets against a further rise in prices, just as the number of U.S. oil rigs in operation hit its highest in over three years.
U.S. shale oil output is up around 10 percent since last year, and, together with increases from the likes of Brazil, threatens to scupper the efforts of the Organization of the Petroleum Exporting Countries and its partners to force a drawdown in global oil inventories via production cuts.
Brent crude futures were up 48 cents at $46.02 per barrel by 0836 GMT. The price is still on track for a near 20 percent drop in the first half of the year, having hit a trough of $44.35 on June 21, its lowest since November.
U.S. West Texas Intermediate crude futures were up 47 cents at $43.48 per barrel.
"We saw this continued big rise in oil rigs last week and in our view we dont need a single additional rig for the next 12 months in the U.S. space if we look at balance for 2018," SEB strategist Bjarne Schieldrop said.
"I dont think OPEC is going to cut deeper, at least not for now. I think its keeping its fingers crossed and looking forward to Q3 and Q4 and hoping their medicine will do the trick."
OPEC and 11 rival exporters agreed in May to extend a 1.8-million-barrels-per-day (bpd) cut in output to March 2018, in the hope that it will force global supply and demand to align.
Traders and analysts said there was little fundamental news to justify more of a bounce in oil prices.
"It is just the fact that the oil market stopped falling ... I suspect short-covering," said Ric Spooner, chief market analyst at CMC Markets in Sydney.
Analysts at Bank of America-Merrill Lynch said that while supply was a problem, demand had not grown quickly enough to mop up any excess output.
"Looking into the second half of 2017, we now doubt that demand growth will accelerate sufficiently," they wrote. (Additional reporting by Jane Chung in SEOUL; Editing by Dale Hudson)