Seven-times European junior individual and team champion Collett, 23, fell from her horse Tis A Beauty while competing at Tweseldown Racecourse in Hampshire on Monday.
"Laura remains in a stable condition but will remain sedated for further 24 hours," read a statement on her official website on Wednesday.
The message from her family continued: "We are all overwhelmed at the response [of well-wishers].”
She awoke on Tuesday and responded to her family before going back to sleep.
A previous British Eventing statement read: "Laura Collett who was competing in Intermediate Section R at Tweseldown Horse Trials One Day Event on the 7th July 2013 has been transferred to hospital after being injured when her horse fell during the cross-country phase.
"The horse, a grey seven year old mare called Tis A Beauty was unharmed."
Collett has been dressage training two-times Cheltenham Gold Cup champion Kauto Star in recent months.
Collett rose to prominence through show pony riding, before switching to eventing and becoming one of Europe's top young riders.
She has since made the step up to the senior squad, competing at the 2011 European Championships, and while she did not make the Olympic team she had been expected to feature prominently for GB over the next few years.