  We prove several new variants of the Lambert series factorization theorem
established in the first article "Generating special arithmetic functions by
Lambert series factorizations" by Merca and Schmidt (2017). Several
characteristic examples of our new results are presented in the article to
motivate the formulations of the generalized factorization theorems.
Applications of these new factorization results include new identities
involving the Euler partition function and the generalized sum-of-divisors
functions, the M\"obius function, Euler's totient function, the Liouville
lambda function, von Mangoldt's lambda function, and the Jordan totient
function.
