The 40-year-old, who is based near Malpas in Cheshire, fell from her mount Topwood Beau and was airlifted to Bristol's Southmead Hospital.
British Eventing (BE) said last week she suffered a traumatic brain injury.
In a further update on Monday, BE said she was about to be moved to the neurological ward.
"Emily is progressing steadily, and the family would like to thank everyone for their messages of support and thank Southmead Intensive Care Unit for their wonderful care," it said.
Topwood Beau, a 14-year-old gelding, was uninjured in the fall, which happened at the third fence - Keepers Question - on the cross-country course.
Gilruth, who is married with two young daughters, made her Badminton debut in last year's event.
She has competed in British Eventing competitions for 18 years and runs her own yard.

Shortly after the show was aired on Sunday, technology entrepreneur Peter Jones tweeted that his investment did not go through.
The Glasgow-based firm had hoped to use the money to help fund the expansion of its Bluetooth tracker, Xupo.
The coin-sized tracking sensor helps people find misplaced personal items.
The tracker was launched by Strathclyde University graduates Raj Sark and Vincenz Klemt in 2014 after they raised nearly £27,500 in a crowdfunding campaign with Kickstarter.
The idea was born when one of the founders lost his travel bag, including a new laptop computer and passport, at Heathrow Airport.
During the show, Peter Jones had offered the investment for 30% equity of the business.
Xupo had forecasted £1m in revenues in the next year.
However, just after the show Peter Jones tweeted: "My investment with My Lupo (sic) didn't go through as some issues arose during due diligence. I still wish Raj all the best in the future."
A PR firm had already issued a press release on behalf of Xupo welcoming the investment.
Mr Sark had said he was "thrilled by the investment and Peter Jones' confidence in the product".
Mr Klemt had said the investment would help them strengthen their position in the UK and expand into Germany and France, followed by Italy and Spain.

The 26-year-old Algeria international has only started two Premier League games this term with a further two starts in a total of 13 appearances.
He is available to make his debut in Tuesday's league game against Burnley.
"We needed another midfield player who can play either side and gives a bit of flexibility," Reds manager Steve Cotterill told BBC Radio Nottingham.
"At the beginning of the week Guy Moussi had a problem with his calf. Even if Guy was fit, at the moment, he's ended up going in the back four for us, which is not ideal. So, I needed another midfield player with some energy in there."
Guedioura initially joined Wolves on loan in January 2010 before signing permanently at the end of the season.
But he has struggled to earn a regular starting place at Molineux and also broke his leg last season.
British rider Emily Gilruth is set to leave intensive care after she was injured in a fall at the Badminton Horse Trials.
[NEXT_CONCEPT]
A tech start-up firm which thought it had secured a £100,000 investment on the BBC's Dragons' Den TV show has had the funding pulled.
[NEXT_CONCEPT]
Nottingham Forest have signed Wolves midfielder Adlene Guedioura on loan until the end of the season.