Starting on the front row alongside polesitter Sergio Perez, Alonso was unable to follow the Mexican in the dominant Red Bull and had to concede another spot to eventual winner Max Verstappen in the sister RB19.
After taking his fourth podium in five grands prix, Alonso conceded his team's perspective on success is starting to shift.
But he is aware that it would take severe Red Bull misfortune for him to finish any higher.
"I think at the beginning of the year a podium was amazing. Now after four podiums we obviously want more, and at least a second place," Alonso said.
"But the two Red Bulls are always unbreakable, and they are always super fast. But as I said maybe Monaco, maybe Barcelona we have a possibility."
Aston's rivals Mercedes and Ferrari both struggled on race pace, leaving Alonso in no man's land to finish 21 seconds behind second-placed Perez and seven seconds ahead of Mercedes' George Russell.
On his cruise to third the two-time world champion even took the time to compliment team-mate Lance Stroll on a tidy overtake he had spotted on the big screens around the Miami International Autodrome.
Source: Autosport