My wife heard this on the radio.

Problem:
The phrase “Second of all,” sounds strained and is otherwise problematic.

Explanation:
The phrase “First of all,” is a popular way to introduce an enumeration because “of all” stresses that one is about to provide an enumeration. For example, First of all, I don’t like beef. stresses to the listener or reader that there will be an enumeration and that what followed “First of all,” was the first item (“I don’t like beef.”) in the enumeration.

Following a use of “First of all,” with a use of “Second of all,”

• sounds strained,
• reduces the power of “First of all,” by making the listener or reader wonder whether every item in the enumeration will include “of all”, and
• does not add anything to saying or writing “Second,”.

Note that it is not appropriate to introduce the second item in an enumeration with the adverbial form “Secondly,” (just as “Firstly,” is not appropriate for beginning the first item in an enumeration).

Solution:
“Second,”