Dartmouth, owned by the Queen, won the Group Three Ormonde Stakes at Chester on Friday.
Jockey Ryan Moore partnered the Sir Michael Stoute-trained favourite to victory from Wicklow Brave and Elidor on his Chester debut.
Dartmouth could now represent Stoute - looking for a 10th Hardwicke Stakes victory - at Royal Ascot in June.
"I should think he's got to dip his toe into a slightly bigger pool now," said the Queen's racing manager John Warren.
"I think the way things are going, he's going to take his chance in an upgrade again."
The four-year-old carried a 3lb penalty into the race as a result of victory in the John Porter Stakes last month.
Jockey Moore added: "He's done well. He's progressed and I'm really pleased with him. It wouldn't really be his track but he enjoyed himself."