The left-sided player had been training with Irn-Bru First Division side St Johnstone but he could not resist Jefferies' invitation to move to Rugby Park.
He said: "I'm delighted. I definitely thought I would be dropping down a division.
"I was on trial with St Johnstone for a couple of days and I was happy to sign for them.
"But when this opportunity came up to join Kilmarnock there wasn't really any decision to be made.
"Every player wants to play in the SPL and if they have the opportunity I think they will take it rather than drop down a division.
"I certainly think I can play in the SPL, I've played a few games for Inverness but circumstances dictated that I was frozen out.
"But there are no hard feelings. Inverness have been a great club for me but hopefully I can kick-start my career with Kilmarnock."
Morgan added: "After the couple of seasons that I've had, in and out the team, playing regularly has got to be my short-term ambition.
"I need to get my match fitness and win a place in the team.
"Jim Jefferies is a well-known figure in football and he is very professional in what he does.
"He gets the best out of players and that is one of the reasons that I am so delighted to sign."
Jefferies welcomed the forthcoming addition to his injury-ravaged squad and thanked Caley player-manager Craig Brewster for tipping him off.
He said: "I'd like to thank Craig Brewster for giving me a courtesy call to inform me that Alan was out of contract and looking for a move back down.
"I watched Alan when he was at Blackburn and I thought he did reasonably well when he got the opportunity at Inverness.
"We are short of left-sided players and we feel that he can be a good addition to our squad."