In the case of this system, the 384k you are referring to is "NOT MAPPED".
That is to say, the normal memory map of a DOS PC looks like this
0kb <--Base Memory-->640kb<--Upper Memory Area / Adapter ROM Region, 384kb of address space-->HMA<-->1MB<--Extended Memory-->
Within the Adapter Rom / Upper Memory Area, you have this breakdown, usually.
Monochrome display adapter ROM and RAM area, (Ram area starts at B000, and ends at B7FF), CGA/EGA/VGA bios area (C000 through CC00 usually, but can go all the way through to CFFF) Then a chunk that is "Free" between D000 and EFFF, but which usually contains disk controller drivers or BIOS. Then, finally the system ROM BIOS at F000-FFFF
The chunk between D000 and EFFF is "Not mapped". That is to say, the addresses there have "Nothing attached to them."
They are reserved for any system option roms you might have/need, and any kind of EMS Ram Solution you might install.
Your 286's memory mapping then likely looks like this:
640k<->Adapter Region<->384kb Extended
with the 384k extended being "Above 1mb", and thus not accessible to DOS.
Your system's chipset might have functions to remap the Extended memory into the unused areas of the Adapter Rom region, but this is very chipset specific. the 286 lacks an MMU, and thus cannot remap the pages directly itself, like a 386 can.
To get UMB there, you need something that can supply RAM in that unused space. Something like a LoTech memory card/EMS card. (which can be configured to just present as UMB in the under 1mb address space.) You will need to tell DOS to turn it on, with something like HIRAM.SYS
The small chunk in the monochrome display buffer area can probably be turned on with hiram.sys with just a VGA card installed. IIRC, it's some 16kb in size?