That is a request from dearieme, and the answer is yes, the case for free trade is still valid.
First, some welfare states, such as the United States and Denmark, are quite compatible with full employment, or could be compatible with full employment if say monetary policy were better. The welfare state may still, through say tax rate effects, keep some family second earners out of the work force. That is likely inefficient, but it doesn’t boost the case for protectionism.
Second, the actual second best problem comes when a welfare state (especially a poorly designed one, and there are some of those) interacts with job churn. Given that some people are out of work, the welfare state may limit their incentive for job search, or the associated taxes and regulations may limit job creation on the employer side. So some workers will lose their jobs due to foreign competition, and find reemployment difficult or not sufficiently desirable relative to the dole.
Overall, though, a lot of those jobs were going to disappear anyway, because of either automation or simply shifts in consumer demand. In that sense free trade is simply the “messenger,” rather than a unique villain. Are jobs more precarious in larger trade zones? I can’t recall seeing a protectionist make that case, instead they simply rely on the superficial observation of the first-order, visible effect, namely that some jobs have gone away for trade-related reasons. The possibility of importing intermediate goods makes many jobs more stable, as do exports. There is no a priori reason to expect free trade to under-perform in this regard.
Free trade still gives an economy more wealth for dealing with transition problems, and it gives workers a better chance of finding a new job somewhere else. To be sure, not all classes or regions of workers will benefit from this dynamism at any point in time. But a welfare state will help protect those workers who do not.
For all of those reasons, the case for free trade is robust to having welfare states.
Alternatively, you might try a “race to the bottom” argument for thinking that free trade and welfare states may interact in counterproductive ways. Let’s say that free trade causes governments to compete to lure or keep business activity. That tends to encourage a social welfare state funded through consumption taxes (not corporate taxes), accompanied by a minimum of regulation. That sounds like an OK enough race to me. I’m not even sure there is a race to the bottom on the regulatory side, but at the very least there are incentives for regulation not to exceed a manageable level, again all to the better.