Confused. Need VERY sensitive headphones.

I know this is a newbie type question, but I really need an answer/help.

So, the problem is, my vintage SONY DEJ-1000 Diskman CD player (which still works perfectly!) puts out so little power. It’s 5 milliwatts at 16 ohms out of the headphone jack. I have to turn it up to over 75% of its volume limit to listen to it normally in a quiet room with my B&W headphones,, which ARE rated for use with an iPhone, even has an optional iPhone cord with the built in mic, along with a normal/plain cord. And with an iPhone, no issues driving them. The portable SONY CD player, which I’m talking about, starts clipping a little, past 75% volume. I NEED a good quality headphone that can accept so little power. Yes, I could buy a portable amp, it has a true optical line out, but that just means more weight/pieces to carry around(not ideal). The whole point IS it’s portability. Any ideas??? I have no idea what brand or model of headphones to buy. BTW, I don’t like overly bright (fatiguing), or overly bassy cans, sorry Dr. Dre. Any ideas with the specs I’m dealing with? Output of 5 milliwatts at 16 ohms? If there are no real options, I guess I’ll have to carry a portable headphone amp. I’d rather not though.

Thank you in advance!! David J.
Also, I’m not adverse to buying in-ear headphones, but honestly would prefer over the ear headphones, if any can be driven by my CD player.
   Back in 1990 it costs me $200.00!! I actually have two of them, one I gave to my dad, who never used it, and he gave it back to me. Thanks dad!!😃😃

Thanks again,
                      David J.
The choice you need to take is the one you don't want.

Get a Fiio headphone amp.
Portability - settle for the orig headset.  It's pretty decent. If you want upgraded headphones you NEED an amp period.
The FiiO A3 is only $58 & 3.25 ounces.
Of course if you are set on portable please justify a 20th century portable CD player in the 21st century.