Essex bowled out the visitors for 118 and 88 inside five sessions to win the game on the second day of the match, with Harmer taking five for 23 in the first innings and seven for 38 in the second to record his fourth 10-wicket haul of his first-class career.
Dan Lawrence and Ravi Bopara hit half-centuries for Essex as they reached 214 in reply to Hampshire's 118.
Warwickshire finished the first day on top against Yorkshire, reducing the hosts to 208 for eight in a rain-affected match at York.
Bowlers Craig Miles and Oliver Hannon-Dalby shared five wickets for the loss of eight runs in the afternoon as Yorkshire slipped from 93 for one to 101 for six.
Miles ended the day with four wickets for 32 while Hannon-Dalby took three for 60.
Yorkshire's Gary Ballance continued his good run of form with 54, including a strong top-order partnership with Will Fraine (42) as David Willey added a useful 46 after coming in at number eight.
A Century from Zak Crawley led a Kent recovery as they scored 309 against Nottinghamshire, who ended the day on 30 without loss in Tunbridge Wells.
After five wickets fell before lunch, Crawley scored 111 and was well supported by Grant Stewart in a seventh-wicket stand that added a century.
In Division Two, Leicestershire's Neil Dexter and Hassan Azad put on an impressive 320-run partnership as the Foxes reached 343 for five at the end of the first day's play against Gloucestershire at the Fischer County Ground.
Dexter hit a career-best 180 while Azad's 137 was the 25-year-old's first Championship century, in only his seventh match.
James Anderson gave another impressive demonstration of his craft with five wickets as Division Two leaders Lancashire took control on a rain-shortened opening day of the match against Derbyshire.
The England fast bowler took five wickets for 18 runs in 17 overs, his most economical in county cricket, while Graham Onions claimed three for 19 in 14 to give Lancashire maximum batting points.
Leus Du Plooy top-scored with an unbeaten 38 out of Derbyshire's 153 and – in reply – the visitors lost Keaton Jennings for a duck to close on eight for one.
Centuries for Paul Stirling and Sam Robson put Middlesex firmly in charge after two days of their clash with Glamorgan at Radlett.
Stirling hit a Championship career-best 138, his first four-day century in almost two years, while Robson's 107 was his first ton of the campaign.
In a quirk of fate, Ireland international Stirling's previous career-best – 111 against Yorkshire in June, 2017 – also came on a day when Robson hit three figures.
The duo's efforts carried the hosts to 410 before the bowlers took over to reduce Glamorgan to 112 for three in an elongated final session.