We discuss the congruences $theta$ that are connected as  elements of the (totally disconnected) congruence frame $CF L$,  and show that they are in a one-to-one correspondence with the completely prime elements of $L$, giving an explicit formula. Then we investigate those frames $L$ with enough connected congruences to cover the whole of $CF L$. They are, among others, shown to be $T_D$-spatial;  characteristics for some special cases (Boolean, linear, scattered and Noetherian) are presented.