Timothy Taylor, at the Conversable Economist, has an awesome graph and an excellent discussion of trade in the modern world.
We often picture some companies producing for export but using as inputs almost entirely domestically produced items. Wrong!
His post is titled "What If US Importers and Exporters are Largely the Same?" If you don't have time to read it, at least let this graph sink in.
The obvious implication: If the feds raise the tariff on imports, then the cost of producing many exports will rise. So, for example, if the feds impose a tariff on steel imports, the cost of producing cars will rise. (Although not as much as you might think because cars have way less steel in them than they used to--but the point remains.)
We've known this for a long time. There was a whole literature, to which my undergraduate International Trade professor at the University of Western Ontario, J. Clark Leith, contributed, on effective rates of protection. This literature dealt with this point. But the literature, if I recall correctly, wasn't so granular. Also, that was over 40 years ago, when trade and international supply chains weren't nearly so extensive. Here are an NBER study (gated, unfortunately) and a published article that are behind this graph.