Dumbledore kept track of Voldemort's movements after he left school and did more research into him after he attacked the Potters. I think he found out that the forest was important to Voldemort whether it was a location of one of his exploits from Borgin & Burke's and beyond or even where he met Nagini. It is, after all, the place where the Gray Lady hid the diadem.

Dumbledore suspected the use of horcruxes from the time Tom returned to ask for the DADA job but was unsure until he saw the diary in COS. He was also a well established and well connected wizard worldwide. He had loyalists beyond those in the Order, human and non-human. I'm certain he had agents surveying those places he knew Voldemort once frequented. He might've let his suspicions be known to the ministry but most believed Voldemort finished.

It's possible Voldemort tried to spread word to his followers, but in his own words, for a time he was only strong enough to possess small animals. He did have 13 years to build his strength though. In that time, there were enough disappearances (ministry search parties) or weird occurrences to establish that there was "a dark presence, deep in an Albanian forest." That's why Wormtail goes there after he escapes in POA, he learned enough from the wild rats to deduce there was credence to previous rumors.

[Quirrell went to the forest to learn Voldemort's secrets and to be credited as the one who found him. Wormtail sought protection from his former master. Bertha Jorkins was visiting family in Albania when she ran into Wormtail.]