On a problem related to "second" best approximations to a real number

For a given irrational number $\alpha$ one can define an irrationality measure function ${\psi }_{\alpha }^{[2]}(t)=\underset{\begin{array}{c}(q,p):q,p\in \mathbb{Z},1⩽q⩽t,\\ (p,q)\mathrm{\ne }(p_n,q_n)\mathrm{\forall }n\in {\mathbb{Z}}_{+}\end{array}}{\min }\mathrm{\mid }q\alpha -p\mathrm{\mid }$, related to the second-best approximations to $\alpha$. In 2017 Moshchevitin studied the corresponding Diophantine constant $\mathfrak{k}(\alpha )=\underset{t\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}t\cdot {\psi }_{\alpha }^{[2]}(t)$ and the corresponding spectrum ${\mathbb{L}}_2=\{\lambda \mathrm{\mid }\mathrm{\exists }\alpha \in \mathbb{R}\setminus \mathbb{Q}\text{\hspace{0.17em}⁣}:\lambda =\mathfrak{k}(\alpha )\}$. In particular, he calculated two largest elements of the spectrum ${\mathbb{L}}_2$. In the present paper we calculate the value for the third element of the spectrum ${\mathbb{L}}_2$.