The 34-year-old joins the county as cover for fellow West Indies paceman Fidel Edwards, who broke an ankle while playing football last week.
Best has played 25 Tests and 26 one-day internationals for his country, and taken 316 wickets in 115 first-class wickets at an average of 28.
Like Edwards, he is a Kolpak signing, with South African Ryan McLaren Hampshire's designated overseas player.
"Tino trained with us in Barbados during pre-season and looked good," Hampshire director of cricket Giles White said.
"It's come about at short notice, initially on a short-term contract. He seems excited about the opportunity and very motivated to do well."
Best will play for Hampshire in a three-day second XI match against Somerset at the Ageas Bowl starting on Monday with a view to joining the first-team squad for the County Championship game that begins against Middlesex on Sunday.
As well as Edwards, who will undergo surgery on Tuesday, Hampshire are currently without the services of bowlers Reece Topley, Gareth Berg and Ryan Stevenson through injury.
Ireland seamer Craig Young has also joined the county on a short-term loan deal.
"We'll support Fidel through his recovery and look forward to having him back in a Hampshire shirt as soon as possible," White added. "He's a class act and the team will miss him."

The 26-year-old Scottish forward was sold to the Ibrox club by Saints for £500,000 in February 2016.
But he found starts hard to come by despite 24 appearances last season and has been sent back to Perth on loan.
"He knows when I need a kick up the backside or an arm round the shoulder," O'Halloran said of Wright.
O'Halloran joined Saints after leaving Bolton Wanderers in January 2014 and, after 16 goals in 90 appearances, was persuaded to move down a division to join Mark Warburton's Rangers.
He scored three goals as he helped the Glasgow side win the Championship title, but he failed to find the net last season and fell out of favour under new manager Pedro Caixinha.
"I came here a few years ago from Bolton and, at the time, it felt right when I spoke to the manager and again it feels right," said O'Halloran of his six-month loan.
"He knows me as a person and how to get the best out of me as a player and I trust him and I think he trusts me, which is important.
"You also see how he is with the boys, it is such a tight group, a family club and we work hard for each other.
"The manager, the staff and the boys were a big part in coming back and I am just looking forward to it."
O'Halloran came on at half-time in Saturday's friendly against East Fife and is hoping to start in this weekend's Premiership opener against Kilmarnock.
"The last game I started was in January," he added. "I haven't done a lot of training sessions and pre-season has been on my own as well.
"It was good to get training with the boys again and get some minutes on the pitch on Saturday, which was important as well going into the season."
Fast bowler Tino Best has joined Hampshire on a short-term deal.
[NEXT_CONCEPT]
Michael O'Halloran believes St Johnstone manager Tommy Wright is the best man to help him rediscover the form that led to a transfer to Rangers.