Hey, I'm glad you like the review.
If you need an open-back headphone that will sound great with all genres of music, but also the bass heavy stuff, in a 150 to 250 dollar range, and you want to run them without an amp, then then you don't have much of a choice, since such headphones don't exist, at least I don't know about them.
The only open-back headphones that you could run without an amp, and not totally compromise their performance, would be Grados and Audio Technicas, but they're not exactly known for their bass performance, if you're used to HD380's, they'll sound very bass light (especially sub-bass which is particularly poor on Grados) and extremely bright to you. In other words, they're not fit for rap/hip-hop or similar genres at all, they're more fit for vocal, acoustic, classical, etc.
Other choices might the open-back Sennheiser HD518 or 558, but they're not exactly easy to run either, they have a huge impedance spike at bass to low mids level that goes up to 300 ohms, which makes them sound bass light without proper amplification.
Now, if you decide to get an amp/DAC, then the best open-back headphones for those bass heavy genres (while still sounding very good with everything else) up to 250 dollars would be Beyerdynamic DT990's and Ultrasone HFI-2400's. 990's are available in either the 250's ohm Pro version or the 990 Premium 32 Ohm version (ignore the 600 ohm version, way too difficult to run), both can be used without an amp without sounding horrible, but depending on what you plug them into now (smartphone, portable player?), you might not get enough volume. In other words, both need an amp and DAC to sound best, but you can get something like the Fiio E10, or anything similar for around 70-80 dollars and it will improve their sound greatly. The same goes for the Ultrasone HFI-2400's. I personally prefer the DT990's because of their comfort and more solid build quality (although both are built like tanks), and a slightly more balanced sound, but both headphones can kick like a mule when it comes to bass, while still sounding very natural, airy, spacious, open and detailed with other music genres. Ultrasone is especially good when it comes to soundstage and imaging, some of the best I've ever heard in a headphone regardless of price.
If you can't get a DAC/amp, or at least an amp now, then I'd recommend you to just stick to HD380's until you can afford to get the whole setup, because otherwise you'll end up buying headphones that will sound worse than what you have now. Open-back headphones are very tricky in this price range, because most of them are very picky when it comes to genres of music they excel at, and most go for the bright approach, which means they really don't do the bass heavy stuff.
EDIT: I totally forgot about one more headphone, the Philips Fidelio X1. That might be the ultimate choice for you. Now, I haven't heard them, but from what I've read about them, they do fit all the boxes. They're well made, have detachable cables, are fairly easy to run and sound great with all genres including the bass heavy ones for what. I can't personally recommend them since I have no experience with them, but they might be the best choice.