After the first two witnesses testified in public, journalists were barred from the room as the third witness, whose identity was concealed, gave his evidence behind closed doors.
In her evidence, Canadian national Caitlan Patterson, who had a two-year affair with Odumbe, claimed alleged Indian bookmaker Jagdish Sodha had visited Kenya in September 1999.
"He came to our house and I made him a cup of tea and he was surprised that I could make very good chai (Swahili word for tea)," Patterson told the tribunal.
Patterson said Sodha used to telephone their house, and that she accompanied Odumbe on trips to Zimbabwe, Namibia and India.
Chief investigator of the ICC Anti-Corruption and Security Unit, Jeffrey Rees, said that all players involved in the last World Cup in South Africa were advised against getting involved with bookmakers and punters.
"History had shown that bookmakers and punters would be interested in the last World Cup. We did not want the competition to be tainted as the previous championship in England in 1999," said Rees, the fourth prosecution witness.
But he said that because of the colossal amount of money involved it had become difficult to eradicate the vice from the sport.
"The extent of betting in cricket is incredible. There is a significant and growing market," Rees said.
Rees, who has served 35 years as a police officer in England, said he had deployed five regional security managers to act as a deterrent, as well as a detection, of match-fixing within the Test and One-day playing nations.
Odumbe, who has denied receiving payments from an Indian bookmaker to fix matches in 2002, faces a minimum two-year suspension or a possible life ban from the game.
The hearing continues Wednesday with more evidence from prosecution witnesses, including an Indian policeman, who has been investigating Odumbe's trips to India.