It's not quite as simple as chlorine has no smell. Chlorine does in fact have a smell even if not reacting with any ammonia in your nose to form monochloramine. The odor threshold for both are fairly low and your nose has chemical detectors for chlorine chemicals, not just for monochloramine. As noted in
this link and
this link, the odor and taste thresholds for various chlorine compounds are as follows:
Chemical .................
Odor .......
Taste
Hypochlorous Acid ... 0.28 mg/L .. 0.24 mg/L
Hypochlorite Ion ...... 0.36 mg/L .. 0.30 mg/L
Monochloramine ...... 0.65 mg/L .. 0.48 mg/L
Dichloramine ........... 0.15 mg/L .. 0.13 mg/L
Nitrogen Trichloride ... 0.02 mg/L ... ?
Chlorinating liquid and bleach have very high hypochlorite ion concentrations (50,000+ mg/L) so it is not at all surprising that they are readily detected in concentrated form though the amount of such gas that can be emitted is limited since it is a charged ion. The concentration of hypochlorous acid in 8.25% bleach is roughly 2.3 mg/L while in 12.5% it is roughly 0.8 mg/L (it's less because the pH is higher at 12.5 compared to 11.9 for the 8.25% bleach). So even if no hypochlorite ion were outgassed, you would still smell the hypochlorous acid, but it wouldn't be nearly as strong as what you currently smell from hypochlorite ion.
In pools at the minimum FC/CYA level (7.5% ratio) at a pH near 7.5 the hypochlorous acid and hypochlorite ion concentrations are each around 0.03 mg/L so would generally not be detected. However,
the above thresholds are for still water whereas churning the water to volatize the chlorine in the water can certainly get concentrations high enough in air that one can faintly smell the chlorine, mostly of hypochlorous acid.
This business of "you can't smell chlorine" and "if you can smell chlorine, then it's really chloramine and you need to handle that in your pool" are extrapolated truths mostly coming from the far more extreme situations in high bather-load pools where higher chloramine levels are really producing more dichloramine and very irritating and smelly nitrogen trichloride. These are the "bad pool smell" associated with improperly managed pools especially those with high bather loads. If the chlorine level is too low in such pools then one can build up organic precursors such as urea or build up monochloramine, but what is more likely to happen is that the chlorine level is then raised and reacts with the urea that has built up and that ends up producing the very nasty nitrogen trichloride. However, if the active chlorine level is too high, then this also produces more nitrogen trichloride from chlorine oxidation of ammonia and related compounds. So too much chlorine isn't good either. Managing this using chlorine alone requires elevating the chlorine level at night when no one is in the pool and aerating the water or at least having strong air currents to remove the disinfection by-products before lowering the chlorine level again for the start of the next day. Using non-chlorine techniques would be better with supplemental oxidation or coagulation/filtration to remove organic precursors.
I think far too much emphasis is placed on "chlorine smell" in residential pools. In some cases where you have higher bather-loads or you have indoor pools with no UV exposure (or UV systems) you can end up with too much urea and get nitrogen trichloride smell, but it doesn't necessarily mean your chlorine level was too low. It just means you are trying to have chlorine oxidize more than it can reasonably handle on its own and one should consider supplemental oxidation systems (e.g. ozone, UV) or enzymes, but for outdoor residential pools these should not normally be necessary. A "clean bleach-like smell" when churning the water in a pool does not mean there is a problem. It's normal. Now when you get out of a pool and you smell your skin then you are likely smelling monochloramine plus maybe some other chloramines because chlorine will continue to react with ammonia from your sweat when you leave the pool and chlorine reacts with ammonia to form monochloramine in under a minute (at our usual FC/CYA levels).