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 want obviously 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 Martin'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.
Alonso admitted he had expected a fiercer battle this weekend as Ferrari and Mercedes couldn't back up their recent improvements in Baku and Melbourne respectively.
"A little bit of a lonely race today and nothing really to do in front of us with the Red Bulls and behind us," he shrugged.
"Maybe we did expect a little bit stronger opposition, but the Ferraris were a bit worse than expected today. So yeah, it's never easy, but it was a lonely race.
"We take the podium and going into the triple-header of Imola, Monaco and Barcelona, at which hopefully we can keep on going."
Alonso has now scored 15 points in every 2023 race weekend so far, missing the podium in Baku but adding to his tally there in 2023's first sprint race.
At 75 points, the Spaniard heads to the Emilia Romagna Grand Prix in two weeks' time with a 19-point advantage on Mercedes' Lewis Hamilton in the battle for third behind the elusive Red Bulls.