A change of command, should really be a change in command. When a senior executive like a CEO to remain in place during an extended transition period it most likely causes conflict and strife. Who is really in charge, the old CEO now chairman, or the new CEO?
In the case where both CEO's are qualified individuals, the organization will split alliances. Any co-leadership creates friction in the command structure that overtime break down the companies ability to successfully compete. In the long run, all business relationships suffer under these conditions.
For the outgoing CEO:
Now is your opportunity to pursue new ventures, look toward hobbies and charitable activities. Is there something you've always wanted to do, but didn't have the time while CEO? Have something to pursue that doesn't involve the old company. Whether golf, volunteer work, or taking up your favorite cause, your separation will be more fulfilling if you embrace these new roles.
It's hard to see your child all grown up, but it's better to let them make their own mistakes. You've invested time and effort into making the company what it is today, the real proof of your effort is how well it can stand without you. Like a child leaving home for their first day of college, this is a test of the lessons you have instilled in them. Use restraint in solving the companies problems after you have left.
Shadow the new CEO in private, be a mentor for the individual over the company. Instead of staying in the companies leadership, be a private mentor to the new CEO, answering their questions and supporting their needs. If they need help share it, but let the new CEO be the figure head for the company.
Discover your new role as a shareholder, or capital investor. You've worked for the company, now it is time for the company to work for you. Attend shareholder meetings and if you feel compelled to stay involved, do it as a concerned shareholder, not former CEO. Your company will be stronger for your actions.
Share your knowledge with a start-up or younger executive in another company. Your knowledge is valuable, but may second-guess the new CEO if shared with your old company. Seek out a small start-up or someone who will truly appreciate your insights and experience. Spend time building a new company, perhaps even another legacy.
For the new CEO:
Encourage the company to clearly define the parting CEO's exit strategy. Specifically define the transition period for the former CEO, make the exit an event, celebrate their contribution to the company. Your exit strategy should include several brainstorming sessions to pass along vital things you must know about the business, and be of a fixed time frame.
Avoid going head-to-head in a power struggle, it will only hurt your career. If a conflict occurs, seek outside support to resolve it. Be respectful of the former CEO's knowledge, but encourage them to pursue other channels. Conflicts will split the company and demonstrate weaknesses in your ability to lead.
If you need to seek guidance or direction from the former CEO, do it in private. Avoid bringing the old CEO back after the transition period, if you need their feedback then go to them in private. Bringing back the former CEO every time the company has a problem undermines your authority. Private sessions are less threatening to you and the former CEO.
Understand the former CEO knows things about the company you haven't yet learned. Maintaining a positive relationship with the former CEO simplifies your ability to led the company in their absence. Respect this knowledge and know when to call upon it.
Encourage the former CEO in whatever it is they are pursuing now. Be excited for the former CEO's opportunity to pursue other ventures, encourage them in their actions. Leaving the familiar is never easy for anyone, be there to assure them the company is in good hands, but you aren't afraid to come for advice when needed.