Note that if there exists a covering system $S$ with modulus $m$, then there is a covering system with modulus $km$ by just taking the $S$ throwing in a redundant restriction mod $km$. We can thus talk about minimal covering numbers, where by a minimal covering number we mean a number $m$ such that there is a covering system where $m$ is the LCM of the system, but where no proper divisor of $m$ is if the LCM of a system. The sequence of minimal covering numbers is A160559 and begins $12$, $80$, $90$, $210$... .

One thing to note which leads to a related question: One can have minimal covering systems with $\frac{\sigma(m)}{m}$ arbitrarily close to 2, and the subsequence of $2^{p-1}p$ is an example of such. This leads to the second question: Is there some non-trivial improvement of this sort of inequality for minimal covering systems? In particular, can we find a slow growing increasing function $f(x)$ such that if $m$ is a minimal covering then $$\sigma(m) \geq 2m\left(1+\frac{1}{f(m)}\right)$$ and where that same inequality is not satisfied by all sufficiently large primitive abundant numbers? There are inequalities in the Sun and Guo paper that are almost of this form, but there one seems to need a function $f(x)$ which takes in input about the prime factorization of $m$ and so isn't an increasing function.

Note that if there exists a covering system $S$ with modulus $m$, then there is a covering system with modulus $km$ by just taking the $S$ throwing in a redundant restriction mod $km$. We can thus talk about minimal covering numbers, where by a minimal covering number we mean a number $m$ such that there is a covering system where $m$ is the LCM of the system, but where no proper divisor of $m$ is if the LCM of a system. The sequence of minimal covering numbers is A160559 and begins $12$, $80$, $90$, $210$... .

One thing to note which leads to a related question: One can have minimal covering systems with $\frac{\sigma(m)}{m}$ arbitrarily close to 2, and the subsequence of $2^{p-1}p$ is an example of such. This leads to the second question: Is there some non-trivial improvement of this sort of inequality for minimal covering systems? In particular, can we find a slow growing increasing function $f(x)$ such that if $m$ is a minimal covering then $$\sigma(m) \geq 2m\left(1+\frac{1}{f(m)}\right)$$ and where that same inequality is not satisfied by all sufficiently large primitive abundant numbers? There are inequalities in the Sun and Guo paper that are almost of this form, but there one seems to need a function $f(x)$ which takes in input about the prime factorization of $m$ and so isn't an increasing function. This question may also be relevant.

Second, there is an open problem of Erdos and Selfridge asking whether there is any covering system with all moduli odd. This is equivalent to asking whether the minimal covering numbers are always even. Here, the size of such an $m$ must be very large under some reasonable assumptions. For example, (3)(5)(7)(11)(13), but there's a paper by Song Guo, Zhi-Wei Sun that shows that an odd square free $m$ must have at least 22 distinct prime divisors. So, it seems like in the hypothetical case of odd $m$, in at least the squarefree situation, we'd expect such an $m$ to be large compared to the smallest squarefree primitive odd abundant numbers.

Second, there is an open problem of Erdos and Selfridge asking whether there is any covering system with all moduli odd. This is equivalent to asking whether the minimal covering numbers are always even. Here, the size of such an $m$ must be very large under some reasonable assumptions. For example, $(3)(5)(7)(11)(13)$ is an odd primitive abundant number but there's a paper by Song Guo, Zhi-Wei Sun that shows that an odd square free $m$ must have at least 22 distinct prime divisors. So, it seems like in the hypothetical case of odd $m$, in at least the squarefree situation, we'd expect such an $m$ to be large compared to the smallest squarefree primitive odd abundant numbers.