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This is a nice question, as it confronts a very replicable and common experience with a well established yet seemingly contradictory fact. As you expected, the smell of metal has nothing to do with the metal actually getting into your nose, as most metals have far too low of a vapor pressure at ordinary temperatures to allow direct detection. The characteristic smell of metal, in fact, is caused by organic substances!

There has been the focus on the specific case of the smell of iron (free-access article!). There are at least two ways in which iron produces a metallic smell. Firstly, acidic substances are capable of corroding iron and steel, releasing phosphorus and carbon atoms present in the metal or alloy. These can react to form volatile organophosphorus compounds such as methylphosphine ($\ce{H3CPH2}$ which have a garlic/metallic odor at small concentrations. From the article:

The “garlic” metallic odor (see Supporting Information) of the gas product from the acidic dissolution of cast iron is dominated by these organophosphines. We measured an extremely low odor threshold for two key odorants, methylphosphine and dimethylphosphine (6 and 3 ng P/m³, respectively, garlic-metallic odor), which belong therefore to the most potent odorants known. Phosphine ($\ce{PH3}$) is not important for this odor because we found it has a much higher odor detection threshold (>10⁶ ng/m³). A “calcium carbide” (or “burned lime”/“cement”) attribute of the general “garlic” odor is probably caused by unsaturated hydrocarbons (alkynes, alkadienes) that are linked to a high carbon content of iron (Table 1, see Supporting Information).

Also, it turns out that $\ce{Fe^{2+}}$ ions (but not $\ce{Fe^{3+}}$) are capable of oxidizing substances present in oils produced by the skin, namely lipid peroxides. A small amount of $\ce{Fe^{2+}}$ ions are produced when iron comes into contact with acids in sweat. These then decompose the oils releasing a mixture of ketones and aldehydes with carbon chains between 6 and 10 atoms long. In particular, most of the smell of metal comes from the unsaturated ketone 1-octen-3-one, which has a fungal/metallic odour even in concentrations as low as $1\ \mu g\ m^{-3}$ . In short:

Sweaty skin corrodes iron metal to form reactive $\ce{Fe^{2+}}$ ions that are oxidized within seconds to $\ce{Fe^{3+}}$ ions while simultaneously reducing and decomposing existing skin lipid peroxides to odorous carbonyl hydrocarbons that are perceived as a metallic odor.

In the supporting information for the article (also free-access), the authors describe experiments performed with other metals, including copper:

Comparison of iron metal with other metals (copper, brass, zinc, etc.): When solid copper metal or brass (copper-zinc alloy) was contacted with the skin instead of iron, a similar metallic odor and GC-peak pattern of carbonyl hydrocarbons was produced and up to one μmole/dm² of monovalent cuprous ion [$\ce{Cu+}$] was detected as a corrosion product (Supporting Figs. S3 to S6). Zinc, a metal that forms $\ce{Zn^{2+}}$ but no stable $\ce{Zn+}$, was hesitant to form metallic odor, except on very strong rubbing of the metal versus skin (that could produce metastable monovalent $\ce{Zn+}$). The use of common color-tests to demonstrate directly on human palm skin the presence of low-valence ions (ferrous and cuprous) from the corrosion of iron, copper and brass alloys is shown in Supporting Figure S6. Alumina powder rubbed on skin did not produce significant odorants. These results provide additional evidence that it is not metal evaporation, but skin lipid peroxide reduction and decomposition by low valence metal ions that produces the odorants.

The last paragraphs of the article summarize the findings:

In conclusion: 1) The typical “musty” metallic odor of iron metal touching skin (epidermis) is caused by volatile carbonyl compounds (aldehydes, ketones) produced through the reaction of skin peroxides with ferrous ions ($\ce{Fe^{2+}}$) that are formed in the sweat-mediated corrosion of iron. $\ce{Fe^{2+}}$ ion containing metal surfaces, rust, drinking water, blood etc., but also copper and brass, give rise to a similar odor on contact with the skin. The human ability to detect this odor is probably a result of the evolutionarily developed but largely dormant ability to smell blood (“blood scent”). 2) The “garlic-carbide” metallic odor of phosphorus- and carbon-rich cast iron and steel under attack by acid, is dominated by volatile organophosphines. Corroding cast iron is an environmental source of C–P compounds that may lead to confusion in the verification and monitoring of the Chemical Weapons Convention (see also ref. [15])

As an aside, this may be why sometimes people recommend getting strong smells off your hands by rubbing them against a metal object. While it probably doesn't work for some metals and for some smelly compounds, it's possible that the metal catalyzes the decomposition of the malodorous substances into less strongly smelling ones.

You can read a little more in this press article on the study.