Tryster is expected to be kept competing at the top table for the time being, with the Juddmonte International at York earmarked as a potential target.
In the hope of encountering a quick surface, the mile-and-a-quarter Group One event on August 17 has been identified by trainer Charlie Appleby as a possible next port of call.
Having registered his first Group One success at Meydan in March, the five-year-old finished fifth on his return to Britain earlier this month in the Prince Of Wales's at Royal Ascot on ground softer than ideal.
Appleby said: "He has come out of Ascot well and I was very pleased with the run given the conditions as he saw it out well.
"We are not using this an excuse but I think he is a better horse on a left-handed track, so I imagine I won't be running him at Ascot or Sandown again.
"We may now look at the Juddmonte for him. I feel that he is better on a flatter track while he wants quick ground. If he gets that he will be competitive at that level."
A visit to the Knavesmire next month could be on the horizon for Wolferton Handicap runner-up Second Wave, with the John Smith's Cup on July 9 coming under consideration.
Appleby added: "Like a lot of the horses that ran at Ascot we have got to see how he comes out of his race, but on first glance he seems like he has taken it well.
"He has had a quiet week and we will now start stepping him up. He is in the John Smith's Cup and he could potentially go for that."
After soft ground ruled impressive Haydock maiden winner Nasimi out of the Empress Stakes at Newmarket on Saturday, a visit to the track could still materialise for a shot at next week's Duchess Of Cambridge Stakes.
Appleby said: "The only reason we didn't run Nasimi at Newmarket was that the ground conditions had deteriorated so badly.
"Although she won her maiden on an easy surface, she is only a slight filly and stepping up in class on that ground is a different level.
"We might now have a look at the Cherry Hinton (Duchess Of Cambridge) with her."