Jo Rugman and Fitzwilliam Square on the course just before the accident.
She was flown to John Radcliffe Hospital in Oxford where she was pronounced dead soon after.
Rugman, of Drayton, Banbury, was following her "A Class Act" team-mates over the tyre fence when her horse, Fitzwilliam Square, stumbled, and clipped the fence.
Rugman's family was on the course at the event, run by the Vale of the White Horse Hunt, which was a qualifier for the National Championships run by the Master of Foxhounds Association.
In team chasing, teams of four riders compete over a cross-country course of about two miles, with about 25 fences to be jumped. The teams set off at intervals and race against the clock. The time of the third member of each team is taken as the time of the team. Team chasing was invented by Hickstead's Douglas Bunn in 1974 as a spectator sport for television, and has had a small but enthusiastic following ever since. The sport has enjoyed a revival in recent years, and has come to public attention by featuring in the BBC Radio series, The Archers.
James Buckle, Chairman of the Masters of Foxhounds Association (MFHA) Team Chasing Committee, said the going on the course was "almost perfect".
"It was a fence unchanged from previous years, being jumped in the middle of the day by one of the most experienced team chasers - what more can one say - a horrendous and cruel accident," he said.
"Jo will be sorely missed by not just team chasing but by the whole hunting community."
The event was abandoned after the accident. Earlier in the day a horse was euthanised after breaking a leg between fences.
Jo Rugman is survived by her husband, John, and two children, Harry (9), and Isabel (7).
Gloucestershire Police are investigating the accident.