He was due to take off on Sunday but was delayed by bad weather.
Mr Piccard will only be able to take short naps while the plane is in flight.
This will be "the longest distance we have had to fly this year," the Solar Impulse team said.
Mr Piccard, a psychiatrist, is sharing the 35,000km (22,000 mile) round-the-world journey with Swiss entrepreneur Andre Borschberg.
The Impulse, which has the wingspan of a Boeing 747, is covered in 17,000 photovoltaic cells to capture the sun`s rays.
It landed at New York`s John F. Kennedy International Airport on June 11 after a five-hour flight from Lehigh Valley in Pennsylvania.
The record attempt began on 9 March 2015 in Abu Dhabi, and has taken the aircraft across Asia and the Pacific to the United States.
LEG 1: 9 March. Abu Dhabi (UAE) to Muscat (Oman) - 772km; 13 Hours 1 Minute
LEG 2: 10 March. Muscat (Oman) to Ahmedabad (India) - 1,593km; 15 Hours 20 Minutes
LEG 3: 18 March. Ahmedabad (India) to Varanasi (India) - 1,170km; 13 Hours 15 Minutes
LEG 4: 18 March. Varanasi (India) to Mandalay (Myanmar) - 1,536km; 13 Hours 29 Minutes
LEG 5: 29 March. Mandalay (Myanmar) to Chongqing (China) - 1,636km; 20 Hours 29 Minutes
LEG 6: 21 April. Chongqing (China) to Nanjing (China) - 1,384km; 17 Hours 22 Minutes
LEG 7: 30 May. Nanjing (China) to Nagoya (Japan) - 2,942km; 1 Day 20 Hours 9 Minutes
LEG 8: 28 June. Nagoya (Japan) to Kalaeloa, Hawaii (US) - 8,924km; 4 Days 21 Hours 52 Minutes
LEG 9: 21 April. Kalaeloa, Hawaii (US) to Mountain View, California (US) - 4,523km; 2 Days 17 Hours 29 Minutes
LEG 10: 2 May. Mountain View, California (US) to Phoenix, Arizona (US) - 1,199km; 15 Hours 52 Minutes
LEG 11: 12 May. Phoenix, Arizona (US) to Tulsa, Oklahoma (US) - 1,570 km; 18 Hours 10 Minutes
LEG 12: 21 May. Tulsa, Oklahoma (US) to Dayton, Ohio (US) - 1,113 km; 16 Hours 34 Minutes
LEG 13: 25 May. Dayton, Ohio (US) to Lehigh Valley, Pennsylvania (US) - 1,044 km; 16 Hours 47 Minutes
LEG 14: 11 June. Lehigh Valley, Pennsylvania (US) to New York (US) - 230km; 4 Hours 41 Minutes
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