Detective Inspector Paul Dalton said the foreign national flew out of Australia on August 31 after driving from Queensland to Sydney by car.
Police intend to charge the man with acts intended to cause grievous bodily harm, which carries a maximum term of life imprisonment.
Dalton said he could not divulge the country where the man was believed to have fled to not jeopardise the investigation.
The baby boy was rushed to hospital after he was scalded at Hanlon Park in Stones Corner about midday on August 27.
The nine-month-old and his mother were having a picnic at the park when they were allegedly confronted by a stranger who poured hot coffee on them.
An off-duty nurse heard the mother’s screams and came to her aid and took her to her home for a shower.
Police have not been able to establish a motive for the attack and the man had not come to the attention of authorities in Australia previously.
“I’d certainly like to know what his motive was,” Dalton said.
“A rational, normal person you think wouldn’t do that but that’s not always the case.
“I’ve seen a lot of disgusting acts.
“I can’t think of someone being in a more vulnerable position than a baby and a mother sitting on the ground.”
Dalton said police only obtained the man’s name on September 1, the day after he left Australia, after initially having been supplied with an incorrect name.
“It soon became apparent to us this person was aware of police methodologies,” Dalton said.
“He was aware what we do to find people.”
The man travelled to Tarragindi and changed his clothes outside a church.
He then caught a rideshare car into central Brisbane, before moving onto Caxton Street.
It’s believed he then travelled to Sydney by car.
Dalton said the man had previously come to Australia legally several times since August 2019 and had several addresses in New South Wales and Victoria.
He was an itinerant worker, he said.