After the Great Depression’s exceptionally high unemployment rates, job prospects did improve in the late 1930’s, but as late as 1941 nearly 1 in 6 people were still out of work.
(A) After the Great Depression’s exceptionally high unemployment rates, job prospects did improve in the late 1930’s, but
(B) Although job prospects did improve in the late 1930’s after the Great Depression, a period in which unemployment rates were exceptionally high,
(C) Following the Great Depression’s exceptionally high unemployment rates, job prospects did improve in the late 1930’s, and
(D) While job prospects did improve in the late 1930’s after the Great Depression’s exceptionally high unemployment rates,
(E) Despite the fact that job prospects did improve in the 1930’s after the Great Depression and its exceptionally high unemployment rates,
Explanation: In (A), (C), and (D) there is a concrete error relating to time that is easily overlooked. You cannot say “after…high unemployment rates” or “following…high unemployment rates.” If you are using “after” or “following” it must be linked with some point in time, action, etc.
It is illogical and incorrect to use rates as a time frame.
Also, the possessive with the Great Depression is a poor structure: you should say the “high unemployment rates of/during the Great Depression” not the “Great Depression’s high unemployment rates” (as if they belonged to the Great Depression).
In (E), “despite the fact that” is wordy and awkward, and more concretely “after the Great Depression AND its exceptionally high unemployment rates” is wrong as it again suggests that something took place after high rates.
Only (B) avoids these problems by showing something took place after the Great Depression (not its rates).
The part of the correct sentence in (B) “a period in which unemployment rates were exceptionally high” is not a structure that many people are comfortable with, but is a common and properly used modifier.