Ok, according to the answers(If I noticed all) I have five variants and questions to them:
Hold your breath (it is not clear then how to remove this air so as not to inhale it after, for example, 10 minutes), it was said by @sonofcy
Buy lead-free solder (however, I do not understand whether rosin is harmful and is it possible to do without a hood?), it was said by @cedarlakeinstruments
Flux smoke tends to goes up. I usually hold my breath until most of the smoke is cleared from around my face, which in my limited experience is typically not more than a few seconds per solder application (I breathe while I check the solder work before I either fix it or do the next solder spot). Holding my breath also helps me be more steady, so it's a win-win.
Airflow doesn't suck directionally, it moves omnidirectionally towards the low pressure. If you want to control a stream of air, you have to make the stream from the source/high pressure side.
Maybe one could duct the exhaust of the filter across the work area and back into the filter
This whole thread is ridiculous! Any lead you may get into your body comes through your skin when you hold the solder in your fingers! The OP will have to wear disposable rubber gloves while handling solder. The fumes are only flux.
I think that to remove smoke it is enough to have a fan, and it should blow not into the soldering area, but into the area of your face, to supply clean air and drive away harmful vapors.
And all these filters do little and are completely unnecessary for DIY. Carbon filters tend to become saturated and need to be changed frequently to work effectively, otherwise they only worsen air circullation.
These recommendations apply to normal living space air and have nothing to do with filtering soldering smoke. It is impossible to say without testing how long your filter lasts for filtering smoke, but my estimate is few days.
Carbon filters in gas masks lasted only a few hours, then they started letting in impurities
The carbon portion of the filter is not an important part of the decision, but a bonus.
The filter unit is only on while soldering, i.e. connected to the soldering iron timer power unit as stated.
In my case, this might be as much as 3-4 hours per week.
As mentioned I only worry about the fumes in the work room; these are removed by the HEPA filter unit.