Roger Charlton could give Projection another try at Group One level in the Darley July Cup at Newmarket on July 14.
The six-furlong contest is one of two races under consideration for the five-year-old, along with the Group Three bet365 Hackwood Stakes at Newbury a week later.
Projection was sent off a 25-1 shot for the Diamond Jubilee Stakes at Royal Ascot but claimed a respectable fifth on what was his first start at Group One level.
Charlton said: “He ran very well at Ascot. He was running out of his class a little bit I thought, but he was only beaten two and a quarter lengths.
“I think his connections would be quite keen to go to the July Cup, but it depends what is going to show up – there are a few that have fallen by the wayside. It’s a very good prize.
“The following weekend there is the Hackwood Stakes and that would be a pretty good alternative at Newbury. He deserves to win a race as he has not won one since he was a two-year-old.”
The Beckhampton handler will look towards an autumn campaign for Aspetar after the the son of Al Kazeem saw his unbeaten run come to an end when trailing home last on his most recent start in a Group Two in France.
He said: “The ground was very firm. We got there and the weather was like it is here now. I walked round and thought ‘this is quick’. I asked Ryan Moore and he said it was good to firm.
“This horse has a knee action and he wants soft ground. I went there as I hoped it would be soft ground, but it didn’t work out.
“We will wait until the autumn and for the ground to come and look at some Listed and Group Three races. It is too early to know if he will be a St Leger horse. He needs to improve a lot, so let’s wait and see.”