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.
               It’s cool, it works, it’s from my “happy time” past. I love it, just as I LOVED my 1968 FIAT Spyder until a very old guy totaled it running thru a red light and crashing into me!!! Was SO mad!! All I have left of it is the wooden and aluminum Nardi steering wheel, the same steering wheel in vintage 60’s Ferrari’s. Has three polished aluminum spokes with slits cut in them, thin wooden rim. Anyway, the point is, it’s portable, I do own iPods and an iPhone as well, but this CD player is special. It’s the best CD Diskman SONY ever made. All magnesium construction, a true optical line-out, 100+ hour battery life, light as hell, lighter than my iPod Classic160 gig. Thin and sexy! And a really cool blue-backlit remote and a nice charging stand. It’s “NOSTALGIA”.
Does that answer your question fuzztone?? Do you have anything from your past you cherish? No idea how old you are. I’m 52. I think a 30 year old, SONY CD Diskman that’s still working is incredible!!That being the issue, totally weak headphone jack output, I’m going to go all in. I’m going to buy a portable amp with a better DAC, than the 1-bit DAC SONY put in there. I assume it’s a SONY DAC, not a Wolffson DAC. Use the line-out to bypass the SONY DAC. Better DAC, portable amp, I’m happy!! I think. Worth a try. I’m not joking, this silver sliver of a CD player makes a hell of a transport in my home system, it would truly amaze you! And because it runs on batteries, no coloration from AC power. It works REALLY well!! Use an AudioQuest Forest optical line-out cable.
Thank you all for your input!!! I’m sad there are no real headphone only solutions, one of you suggested the original headphones, but I threw them away as they sounded awful. But maybe this will be better because I can bypass the OEM DAC with a current, really good one, as well as amplification!