To decide to look for another employer makes sense when in the preceding phase (up to chapter 12) it becomes clear to you that the employer will not or cannot make sufficient choices to your needs you so freshly realized, and that you cannot change enough to have a long and happy future with the same employer.

Then it is good to take your burnout lessons as a map to select potential next employers.

With you current employer, it is best to agree about 'outplacement'. Usually the pressure is taken off you, and you can spend 1-2 days during working time looking elsewhere, with an outplacement consultant.

He should know about the recent burnout.

New, to be approached employers do not need to know about your recent burnout if you are sure you learned your lessons well, and are likely to perform satisfactorily in a new environment that better fits your needs.