The six-year-old gelding showed up well for a long way in the Kingwell Hurdle before he was outmuscled by Melodic Rendezvous and Zarkandar close home.
King was especially encouraged as the Grade One-winning Grumeti is not suited by heavy ground.
Connections will now chart a route towards the Cheltenham Festival for what is shaping up to be an epic renewal of the Champion Hurdle on March 11.
King said: “Grumeti will definitely go for the Champion Hurdle, having run well to finish third in the Kingwell at Wincanton.
“The ground would have been plenty deep enough for him, but he travelled strongly into the race and I would like to think there is still more improvement to come.”
King’s Cheltenham portfolio could be further strengthened by Medermit and Wilde Blue Yonder.
The former finished fourth in the Ascot Chase on Saturday on his first start since April 2012, while Wilde Blue Yonder suffered another last-flight fall in a good novices’ hurdle on the same card.
King said: “Medermit came back fine after his run against Captain Chris in the Ascot Chase.
“Having been off almost two years, I thought he ran really well.
“It was a big ask in that ground in such a hot race, but we needed to get him started again. He is in the Ryanair and the Gold Cup at Cheltenham, or we could wait for Aintree.
“It was tough seeing Wilde Blue Yonder take another last-flight fall at Ascot when about to win his race, but, having looked as if he was on a long stride, he changed his mind.
“Happily, he returned unscathed, and the plan is to still go for the Supreme Novices’ Hurdle at Cheltenham.”
* Cloudy Too is likely to have a crack at the Betfred Cheltenham Gold Cup after his fine effort at Ascot last week.
Trained by Grand National-winning handler Sue Smith, the eight-year-old ran well for a long way in the Hennessy before winning the Rowland Meyrick Chase at Wetherby over Christmas.
He then tackled last week’s Betfair Ascot Chase, where he gave Captain Chris more of a race than anything else before being left in the home straight.
Smith’s husband and assistant Harvey said: “My gut feeling is that he’ll go for the Gold Cup.
“We may as well sink or swim.
“If he runs well then the handicapper will have him right, but if he doesn’t then he will be forced to reassess him.
“I’m looking for a big run from him.”
The Smiths also have a live candidate for the Albert Bartlett Novices’ Hurdle in Blakemount, just touched off in the River Don at Doncaster.
“He’s a nice horse, but he’s still a baby on the way up,” said Smith.
“Horses sometimes need an easy race after a hard one but he’s had lots of hard races.
“He was just touched off on the line at Doncaster and you can accept that, that’s racing.
“He’s very tough and is a nice horse but you’ve got to nurture them.”
* Former Irish Cesarewitch winner Minsk has died at the age of six after contracting a rare illness.
The son of Dalakhani, trained on the Curragh by Dessie Hughes for jumps and by John Oxx for the Flat, fell ill on Sunday and had to be put down shortly afterwards.
Trainer Dessie Hughes explained: “He got an infection in his system and there was nothing they could do. It’s just one of those things.”
Owner Barry Connell paid tribute to his four-time winner and expressed his frustration over the fact he was never able to fulfill his potential.
He said: “He just got very ill. He came down with a form of travel sickness and injured himself and had to be put down. The vets were saying it’s quite rare and that they only get one or two cases each year.
“It seems like he’s been around for a long time because he won the Irish Cesarewitch as a three-year-old, but he’s only six. He went over hurdles after winning the Irish Cesarewitch and I think he was one of the ante-post favourites for the Triumph. We had a very wet winter then and his wind wasn’t 100 per cent so it was always catching him. He probably didn’t ever fulfil his true potential.”
Minsk amassed almost £90,000 in a 20-race career, winning four races including the Irish Cesarewitch in 2011. He won one hurdle race at Punchestown and finished fifth in last year’s Neptune Investment Management Novices’ Hurdle.
* This year’s Investec Oaks will be run in memory of Henry Cecil, Epsom has announced.
Cecil saddled eight winners of the race between 1985 and 2007 and with the 2014 renewal being the first since the trainer’s death last year, officials have decided to pay tribute by running the race as the Investec Oaks (in memory of Henry Cecil) for one year.
Oh So Sharp was Cecil’s first winner in 1985 followed by Diminuendo (1988), Snow Bride (1989), Lady Carla (1996), Reams of Verse (1997), Ramruma (1999), Love Divine (2000) and Light Shift (2007).