He had been granted electronic bail by the courts in January.
Mr Olsen's last known location was Gisborne on Friday, May 22 and police want to ascertain his movements between that date and Tuesday, June 2.
Police are particularly interested in information about how Mr Olsen got to Taupo.
He has links to Wellington and police say they can't rule out that he travelled somewhere else during the 10-day period before reaching Taupo.
"There is no suggestion he had a vehicle, so it is possible he hitch-hiked and police want to speak to anyone who gave him a lift during this period," said a police spokeswoman.
Inquiries are also being made with transport companies.
Mr Olsen is described as part Maori, medium build and approximately 178cm tall with receding dark hair, sometimes tied back.
"Whilst there is nothing at this stage to indicate his death is suspicious; it is important for police to keep an open mind until more details are known of his movements in the week before he was discovered in the river."
If anyone has any information that might assist police inquiries into Mr Olsen's death, they are asked to contact Taupo police on (07) 378 6060.
Information can also be provided anonymously to the organisation Crimestoppers on 0800 555111.
