By a hamiltonian coloring of a connected graph $G$ of order $n\ge 1$ we mean a mapping $c$ of $V\left(G\right)$ into the set of all positive integers such that $|c\left(x\right)-c\left(y\right)|\ge n-1-D_G(x,y)$ (where $D_G(x,y)$ denotes the length of a longest $x-y$ path in $G$) for all distinct $x,y\in G$. In this paper we study hamiltonian colorings of non-hamiltonian connected graphs with long cycles, mainly of connected graphs of order $n\ge 5$ with circumference $n-2$.