末に: a result reached after a lengthy process
Compare the weight of a process in 末に with the simple time sequence in 後で.
Show the process leading up to the result
末に describes reaching a final result after a process such as lengthy discussion or deliberation. 長い議論の末に、全員が案に賛成しました。 means that after a long discussion, everyone agreed to the proposal.
The important point is the process leading to the result. Beyond merely saying which of two events came first, it suggests that reaching the decision took time or involved several stages.
Use の after a noun and the た-form after a verb
Use 議論の末に with a noun and 何度も話し合った末に with a verb. 何度も話し合った末に、新しい方針を決めました。 means that a new policy was decided on after repeated discussions.
The outcome of a long process can be abandoning an option as well as adopting it. よく考えた末に、申し込みをやめました。 means that the speaker decided not to apply after careful thought. 末に does not itself determine success or failure.
When 後で changes the emphasis
昼ご飯を食べた後で、散歩しました。 naturally reports the sequence of eating lunch and then taking a walk. Using 末に in this ordinary setting would present the meal as a lengthy, significant process instead.
Choose 考えた末に when the extent of deliberation matters, or 考えた後で when only the sequence after thinking matters. Both can precede a result, but they do not give the preceding process the same weight.
What does 末に add in 何度も話し合った末に、新しい方針を決めました。?
Show the answer and why
It adds that the policy was finally decided on through repeated discussion, not merely that discussion happened before the decision. 末に does not itself judge whether the decision was good or bad.