Optimal algorithm to traverse all paths in the order of shortest path
The goal is to find all paths from $s$ to $t$ in a directed, acyclic graph with edge costs. Let $n$ the number of nodes, $m$ the number of edges and $p$ the number of $s$-$t$-paths. Let's look at what we have.

Naive enumeration


Use any graph traversal from $s$ to find all paths.
Sort them by length.


You can interleave the steps, but you'll end up with runtime $\Omega(n + p \log p)$ if there are $n$ nodes and $p$ paths.

Note that this algorithm is easy to parallelise.

Eppstein's Algorithm [1]

This one has runtime $O(m + n \log n + p)$ (from skimming the paper, I assume the bound is sharp, i.e. $\Theta$ holds). Note that paths are not represented explicitly.

I have no idea how well this algorithm can be parallelised.

Lower bounds

First of all, $p \in \Omega(2^n)$ in the worst case even in DAGs, see for instance this class of graphs:


[source]

Note that all nodes have small out-degree and all paths have the same length.

Therefore, $p$ dominates any algorithm for this problem (in the worst case) if we can not make further assumptions.

When you go for a traversal over all paths, you clearly have runtime $\Omega(p)$.

Conclusion

Eppstein's algorithm is asymptotically optimal even in the setting you describe. However, asymptotic analysis at this level is rough and hides "extra" work that may or may not pay off in practice (since $n$ does not tend to infinity). The naive version may very well be competitive in an application, depending on $n$, $m$ and $p$, and maybe even structural features.

Another thing to keep in mind is that above figures are all worst-case but you are probably after average runtime on some specific input set².

TCS papers are unlikely to contain a rigorous analysis that helps you decide (and it's hard to do, too). Your question reads as if your goal is a real-world application, so I recommend a pragmatic approach. If the naive approach is too slow for you, try parallelising it. If that is still to slow, you can invest the time, implement Eppstein and run some tests.




Finding the k Shortest Paths by D. Eppstein (1998) [preprint]
My group developed a method for semi-automated average-case analysis and a tool to perform it. Have a look, maybe it can help you find out more about performance on your data.