Thanks for the answers! From the side of the PoE (power over ethernet) hat, I wrote to one of the brand supplying such hats and they told me that indeed PoE hats don't add a WoL capacity to the Pi. Of course, if you have a PoE switch and plug your Pi on it with the help of a PoE hat, then if you turn the switch on/off, you can control the Pi's energy supply and knd of wake it on lan, without the "magic packet".
Yet beware that the solution consisting in adding an ethernet controlled relay, althouch it certainly works to remotely control the Pi, probably drains more power than the Pi itself. I couldn't find precise information, but simple relays like this one (https://www.antratek.nl/eth002) need an AC adapter supplying at least 500 mA with 12V DC, so that would mean a consumption of 6 W, which is twice the consumption of the Pi4 in iddle regime. So good to remotely control the Pi, but not to reduce its consumption.
Finally, wrt my question as to the possibility of using a PoE hat to add the WoL capacity, while still using the AC adapter to supply the power, the answer is no since PoE hats do not add the WoL capability. I don't know what the answer would have been if PoE hats would indeed offer this capability... but they seem not to. Basically, they only make sure that the roughly 48V supply from the Ethernet power cable supporting PoE is transformed to the Pi's requirements (5V and 3A I think...) and protect it from voltage variations or the such...