Can a food contain 'extra' sodium?

I think you've misunderstood people talking about sodium; I would expect they are generally just talking about table salt, NaCl.

Addressing your question about quantity, it's very common that when cooking, people judge ingredients by eye and by their expectation of what a normal quantity would be, so use much less salt than a restaurant or commercial preparation. (Something similar is true of butter.) Commercial ice cream, for example, will contain salt, but home ice cream makers are unlikely to include much if any.

If you want to think of that as 'hiding' sodium in products, I suppose that's reasonable, but it's not being done through mystery sodium compounds you're not aware of.

I too think you have mis-understood what happens when the sodium is declared on the product nutritional information.

To get the sodium content (and other contents too), the food-stuff undergoes a chemical assay that measures the quantity of whatever they are looking for. This will be done in an analytical laboratory using specialist equipment that can measure very accurately the components in the food. I found a technical abstract comparing a few different methods here, which indicates that the methods of choice are one of the variants of mass-spectrometry (MS, how a molecule flies under an ionizing field), spectrophotometry (how much light absorbed by a substance (in this case the sodium which has been ionized by a flame)) or an ion-selective electrode (electrical potential of the ion in solution).

In theory you could add up the sodium content of the components of the food and work out the amount of sodium that is there, however this would be inaccurate because foodstuffs often contain sodium by themselves, without the addition of extra salt containing substances. A fine example of this is the seaweed Nori (the one used around sushi rolls) has about 48 milligrams of sodium per 100 grams naturally, but a seasoned nori can have up to about 2300 mg sodium/100g.

Because of the assays there is no "hiding" of the sodium content in the ingredients, all of the sodium that is there is measured in the assays, no matter if it comes naturally in the food, or is added in the form of sodium chloride, MSG etc.

The answer is "no", but probably not for the reasons you imagined it.

The most widespread mental model on taste is what I would all "analytical": salt ions hit the receptors on your tongue, your brain notices the signal and you say to yourself, "this tastes salty", and this is taste.

I have the impression that the above model is the one most widespread in contemporary Western culture, but it does not actually explain well the realities of taste perception. This is why modern books on the physiology and psychology of taste usually apply a model more congruent with gestalt psychology: when you taste the food, your brain scrambles all signals coming from all sources at once, and the resulting internal experience is the taste. It is unique to the food you are eating, and trying to divide it up is meaningless.

I understand your question, with the update, to state that eating food with sodium and thinking "salty" is based on an implicit "normal" amount of sodium, and asking if adding more sodium will also cause our flavor experience to be changed without us being able to say that it is due to salt.

If you use the first model (which I don't recommend, since it describes reality poorly), it is possible to speak of "flavor enhancers" in the sense of this question. Remember, under this model, aspects of the taste experience which are consciously matched 1:1 to a chemical interaction on the tongue are the norm, and anything else shouldn't exist, or is at least a phenomenon requiring special explanation. This would match your description of "hidden" taste, which sounds a lot like the concept of salt "enhancing" taste, without your consciousness saying "this tastes so-and-so because of salt".

Under this first model, the opposite of your premise occurs. If you don't notice the effects of salt on a food, this is usually the food contains less salt than the amount that becomes salient. You might not recognize the salt in a slice of bread, but you will sure notice that brined herring is salty. This means that the flavor enhancing effect only happens when the food contains small amounts of salt, not large ones. Thus it is not "extra" salt on top of the salt you notice is there. If they were to add extra salt to salty food, you will still notice the salt, and it won't be hidden.

Under the second model, the question becomes meaningless. Since in this model, anything in your food is a contributor to the unique whole experience of taste, being an "enhancer" is the norm for all things in the food. A food containing so much salt that you also makes you recognize its contribution and say "oh, this is salty" is something that happens occasionally, and is possibly more an artefact of how we have learned to think of food (a normal person has learned to associate a taste with a particular profile with salt, while a wine taster has also learned to associate a different flavor profile with berries and notice when it is present or absent in wine). The contribution of recognized and unrecognized aspects to the whole of taste is not qualitatively different, and the attributions we make are not necessarily a true description of what is going on chemically on the tongue. So, declaring part of the taste a consequence of "normal" salt and another part the consequence of "extra" makes no sense. Since the concept of "extra" salt is meaningless, it obviously cannot be found in food, be it from industrial sources or otherwise.