The authors in a previous paper devised certain subcones of the semidefinite plus nonnegative cone and showed that satisfaction of the requirements for membership of those subcones can be detected by solving linear optimization problems (LPs) with $O(n)$ variables and $O(n^2)$ constraints. They also devised LP-based algorithms for testing copositivity using the subcones. In this paper, they investigate the properties of the subcones in more detail and explore larger subcones of the positive semidefinite plus nonnegative cone whose satisfaction of the requirements for membership can be detected by solving LPs. They introduce a {\em semidefinite basis (SD basis)} that is a basis of the space of $n \times n$ symmetric matrices consisting of $n(n+1)/2$ symmetric semidefinite matrices. Using the SD basis, they devise two new subcones for which detection can be done by solving LPs with $O(n^2)$ variables and $O(n^2)$ constraints. The new subcones are larger than the ones in the previous paper and inherit their nice properties. The authors also examine the efficiency of those subcones in numerical experiments. The results show that the subcones are promising for testing copositivity as a useful application.