Morse's promotion fills the gap left by Clare Hornby's departure earlier this year.
A former Campaign Face to Watch, Morse joined the agency from St Luke's in 2004 as the account director on Pot Noodle, COI and The National Magazine Company. She will retain her responsibilities on all three accounts in addition to her new duties as the new-business director.
Cairns, who is also the group account director on the Sky business, joined the agency in 2000. She began her career at BMP DDB before moving to M&C Saatchi.
Cobbold has spent the past 18 months at HHCL as its head of account management.
He has led four of the agency's main accounts: Birds Eye, COI, Georgia Pacific and Alfa Romeo.
The HHCL managing director, Sid McGrath, said: "We have some fantastic talent in the agency and I want to give Kate, Rhona and Jason the opportunity to play a more significant role."