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The complement $M\setminus L$ of the Lagrange spectrum $L$ in the Markovspectrum $M$ was studied by many authors (including Freiman, Berstein, Cusickand Flahive). After their works, we disposed of a countable collection ofpoints in $M\setminus L$.  In this article, we describe the structure of $M\setminus L$ near anon-isolated point $\alpha_{\infty}$ found by Freiman in 1973, and we use thisdescription to exhibit a concrete Cantor set $X$ whose Hausdorff dimensioncoincides with the Hausdorff dimension of $M\setminus L$ near$\alpha_{\infty}$.  A consequence of our results is the lower bound $HD(M\setminus L)>0.353$ onthe Hausdorff dimension $HD(M\setminus L)$ of $M\setminus L$. Anotherby-product of our analysis is the explicit construction of new elements of$M\setminus L$, including its largest known member $c\in M\setminus L$(surpassing the former largest known number $\alpha_4\in M\setminus L$ obtainedby Cusick and Flahive in 1989).