Minimally extend a tree such that there are no bridges in the new graph
Start by proving the following statement:

Let $G=(V,E)$ be an undirected graph. There are no $\mathrm{bridges}$ if and only if there are two edge disjoint spanning trees of $G$.

Proof:

Suppose there are two edge disjoint spanning trees, $T_1, T_2$ of $G$.
Now we remove an arbitrary edge. If it belongs to neither $T_1$, or $T_2$, we're done. If it belongs to $T_1$, it is therefore not in $T_2 $ (symmetrical for $T_2$)  $\Rightarrow$ $G$ is still connected $\Rightarrow$ there are no $\mathrm{bridges}$ in $G$.

Suppose there are no two edge disjoint spanning trees. Then there is at least one cut $C^*$ of $G$ s.t $C^*=\{A, \bar{A}\}$ and $\delta(C^*) \leq 1$. Otherwise it is easy to constructively show there are two spanning trees.
Let $e$ denote the edge between $A$ and $\bar{A}$. Since $e$ is the only edge connecting the two subsets, by removing it, $G$ is separated into two unconnected componenets, therefore $e$ is a $\mathrm{bridge}$



If you can add any edge you want, simply extend $G$ s.t there are two edge disjoint spanning trees. If you start from a tree, it means you have exactly $n-1$ edges, and to create another distinct spanning tree you would have at least $n-1$ additional edges.
If you can't add any edge you want, see http://i.stanford.edu/pub/cstr/reports/cs/tr/74/455/CS-TR-74-455.pdf by Tarjan on disjoint spanning trees (with algorithm)