Rudd held a number of ministerial roles, most recently Secretary of State for Work and Pensions from November 2018 until September 2019, when she resigned from the Conservative Party over inaction on Brexit.
Rudd was MP for Hastings and Rye for almost a decade, standing down ahead of last year's general election.
Her previous roles include Minister for Women and Equalities, where Rudd led on delivering the UK’s first gender pay gap reporting regime.
As Secretary of State for Energy and Climate Change, she led the UK team to the successful completion of the Paris Climate Change Agreement in December 2015.
The former Government whip, Rudd was Home Secretary from July 2016 until April 2018, during which time she responded to a series of terrorist attacks. Rudd also called for an international response to take radicalising material off the internet.
Rudd's brother, Roland, is CEO of Finsbury – a rival of Teneo.
In her letter resigning from Boris Johnson's cabinet, Rudd called the expulsion of 21 MPs from the parliamentary Conservative party over their rebellion of a bill aimed at avoiding no-deal Brexit an "act of political vandalism".
Her letter said: "I joined your cabinet in good faith: accepting that no-deal had to be on the table, because it was the means by which we would have the best chance of achieving a new deal to leave on 31 October.
"However, I no longer believe leaving with a deal is the government's main objective."
She joins a number of former senior politicians as senior advisors at Teneo, including former Conservative Party leader Lord Hague and ex-US Middle East peace envoy George Mitchell. Last year Sally Osman, former comms director at the Royal Household, joined the senior advisory group.
Tempest-Hay's CEO responsibilities will be taken over by Teneo International chairman Charles Watson with support from Teneo’s London strategy and communications chair, Fraser Hardie, and the London leadership team.
That followed news last June that Teneo had secured a new private equity backer: CVC Capital Partners.
Teneo removed the sub-branding from all its global operations in late 2018, making Teneo the name used for the business worldwide with the phasing out of established agency monikers - including Blue Rubicon in the UK.
The agency is listed joint fourth in PRWeek's most recent UK Top 150 Consultancies table, with estimated UK revenue of £45m in 2018.