The All Australian small forward had scans on Monday morning after going down during the last quarter of Sunday's three-point win over North Melbourne.
Fronting the media early on Monday afternoon, the disappointed 23-year-old said he would take a cautious approach in his recovery, having experienced persistent hamstring issues in the past.
"We sat down with the staff today, and we haven't put a timeframe on [a return to the field]," Rioli said.
"I'm thinking it won't be before the bye.
"I'll just get it right."
It is the fourth time since the start of his AFL career that Rioli has strained his right hamstring, having missed three matches in 2009 and a further two blocks of games in 2011.
He also tore his left hamstring in the elimination final loss to Fremantle at the end of 2010.
The livewire's persistent problems forced him to change his running style in mid-2011.
Rioli put the injury down to fatigue, rather than a breakdown in his new technique.
"Talking about the midfield, and getting in there, I think my body got too fatigued, and the hammy just went," he said.
That assessment is sure to raise questions about Rioli's future as a midfielder, with Hawks coach Alastair Clarkson questioning his young star's ability to cope with the midfield workload after his 2011 hamstring tear.
The injury won't require surgery.
Rioli was the star of the victory over the Kangaroos, igniting his team with four goals.
Half-back Grant Birchall said the Hawks would do their best to replace him.
"He's obviously very explosive and a very damaging player," Birchall said.
"He's pretty hard to replace with his skill level and his ball winning ability and his goalkicking and all that sort of stuff, so he'll be sorely missed.
"But we've got some good depth, so I'm confident whoever comes in can fill the void pretty well."
After the Crows, Hawthorn will face the Sydney Swans in a 2012 Grand Final rematch at the MCG before clashes with Greater Western Sydney, Gold Coast and Melbourne leading into the bye.
