Welshman Compton, 27, joins on a one-year deal after leaving Yeovil Town.
Rigg has signed a two-year deal after helping AFC Wimbledon to League Two promotion, playing 44 times last term.
"The squad is starting to shape up nicely. There are a couple more to come in," manager Warren Feeney said.
"I'm very happy with the signings we have been able to secure so far."
Former West Bromwich Albion player Compton started his career with Portsmouth and joined the south coast club again for a short spell in 2012.
Compton also featured for Bradford, Colchester and Hartlepool United in addition to a spell in Scotland with Falkirk.
Last season he scored four goals in 20 appearances for Yeovil, including the winning goal in the Glovers' 1-0 win over the Exiles in April.
Rigg, also 27, started at Newport's rivals Bristol Rovers before spells with Port Vale and Oxford United and scored twice for the Dons last season.
"I'm delighted to have signed for Newport County," Rigg said. "I took to Warren Feeney straight away and that was a big draw in me signing."
In a summer where several players have departed Rodney Parade, Jamie Turley, Scot Bennett, Mark Randall, Joss Labadie Jennison Myrie-Williams and James Bittner have all been recruited.

The college, which employs about 480 staff, said it had been operating with a "staff surplus" for several years which must now be addressed.
The decision "has no impact" on students, it said, and although 76 posts might go, it was anticipated the final figure "will be less than this".
UCU (University and College Union) said it had "seen it coming to some extent" but was "shocked" at the scale of it.
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The college said it would be offering voluntary redundancies to reduce the need for compulsory redundancies.
Interim and agency staffing would be "significantly" reduced and there would be opportunities for staff to retrain, it added.
A 30-day consultation has started to discuss the proposals.
College principal and chief executive Graham Guest said its priority was to "build on the improvements noted and highlighted by both Ofsted and the FE commissioner" during recent visits.
In its June 2016 inspection, Ofsted found the college was operating inadequately but has found improvements during two recent visits.
The college is in the process of merging with the town's New College, but said the job proposals were nothing to do with that.
It could not say if there would be further job losses when the merger takes place.
Vice chair of the college branch of UCU Crispin Barker said the union had been in communication with management and knew about the financial pressure the college was under.
Mr Barker said he was "sure" the figure of 76 would be reduced from what he had been told by the college.
The union said it intended to work with college leaders to minimise any "potential impact upon the learner experience".

Falcons lie eighth, having won six of their 14 matches, after finishing 11th for the previous three campaigns.
"Every year I've been here we've shown improvement," the 53-year-old told BBC Radio Newcastle.
"People don't quite see improvement though until you overtake people, which is what we've done this year," he said.
"I've never looked at anything other than that [being in the top six] for the last four years. We're now getting more wins than people thought, and people are now considering us as top six contenders."
Newcastle host Northampton, who they beat 22-16 earlier this season, on Sunday, with the Saints one place above the Falcons in the table.
"They fight for every point they can get, and will try to reverse the defeat from earlier in the year."
"I want people to look at our combativeness and 'never willing to lie down' attitude. We are getting there. Every game has been a battle for other teams. They don't like coming here and we know that. It's not about the weather up here, but the way we play."
"We've still got to go out and do it. It's not about dizzy heights and aspirations. It's about nailing that win time and time again."
Newport County have signed winger Jack Compton and striker Sean Rigg.
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More than 70 jobs are facing the axe at Telford College of Arts and Technology.
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Newcastle director of rugby Dean Richards believes his side are gaining attention due to their higher position in the Premiership table.