Target the transitioning on the Anchor elements , and add an default opacity for the non-hover state.
.contact-links a {
opacity:1;
position:absolute;
float:left; -webkit-transition:all .3s ease-in-out;
-moz-transition:all .3s ease-in-out;
-ms-transition:all .3s ease-in-out;
-o-transition:all .3s ease-in-out;
transition:all .3s ease-in-out; }