About posets for which no upper cover or no lower cover has the fixed point property

For a finite non-empty set $X$, let $\mathfrak{P}(X)$ denote the set of all posets with carrier $X$, ordered by inclusion of their partial order relations. We investigate properties of posets $P \in \mathfrak{P}(X)$ for which all upper covers or all lower covers in $\mathfrak{P}(X)$ do not have the fixed point property. We derive two conditions, one of them sufficient for that no upper cover of $P$ has the fixed point property, the other one sufficient for that no lower cover of $P$ has the fixed point property, and we apply these results on several types of posets.