For a graph $G$ find the minimal $t$ such $G(t)=(V,E(t))$ is connected
Compute a minimum bottleneck spanning tree of the graph. (Actually just compute a minimum spanning tree, as a minimum spanning tree is a minimum bottleneck tree.)
Determine the maximum edge weight from MBST, this will be your $t$.
Remove all edges whose weight is higher.


This can be done in linear time in the sum of the number of nodes and edges (see wiki).