Yeah, I've played with choosing different amps... It's just odd.
I'm going to play around some this weekend and see if I can articulate better. It's just sort of sad when the sound is harsher on my ears than the Voice Memo app on my phone.
*edit* I didn't even look at who I was responding to. That being said..bygons.. as far as I'm concerned we're all fellow musicians as well as human beings and I'm leaving it at that.
Two words. "Impulse Responses". Don't get me wrong, there's a lot more that goes into a good recording and mastering of a bass signal. But recent experiences have pointed me towards cabinet simulations as having a HUGELY positive effect on my bass's tone through headphones from an audio interface.
Strings are an option but hear me out here. My P-bass had Rotosound 77 Jazz flats .45-.105on it for about a year and frankly, they sounded freaking great. They were stiff as all heck. But that bass didn't need EQ. I could plug it direct into my interface and it would produce sweet sweet tones. I could barely hear the difference in rolling the tone on and off unless I was playing with a backing track and other instruments. Silky smooth.. you maybe barely heard the occasional clack of a fret or my finger grazing a string with the tone full on.. rolled off it was just eliminating the string noise.
I decided in like August or September to see why people on these forums rave about TI Flats. They're a thinner gauge and boy they freaking sound like it too. Plugged directly into my interface that bass sounded like

in a tin can. The tone thickened up when I rolled the tone knob off but the more tone I added the more cringeworthy it got. As I've seen another member say..it was like the bass had lost all it's booty. The following weekend I took my P-bass to my band's jamspace where my 410 cabinet lived. I plug into my amp and the bass just sounds fantastic. I was relieved, but confused. So I take my amp home with me (not my cab) and run a post-EQ signal from the XLR out. The EQ helps.. but it's still not the wonderful tone I was getting from my amp and cabinet.
Fast forward a few months I get a Pod Go for my birthday... I have a Strat and no guitar gear. A month and a half later I'm still not playing my Strat much but I use the Pod with my bass daily. It has several bass amp or just the preamps of those amps as well as several bass cabinet simulations.. I also 4 cable routed the Pod through the effects loop of my amp. I can switch between my amp and any given one of the Pod's amp simulations by toggling a footswitch.
The amp sims on their own.. while they have their own flavors, as well as my amp having it's own characteristics. I'm able to get similar tones between a few of them however, I've even gotten some of those amp sims pretty close to my amp with the right settings on both. But those cabinet simulations man.. the SVT 810 Sim is like instant booty. DEEP lows that sound so good. So now the TI Flats not only feel good under my fingers but sound good in my recording setup. Other cab sims might sound a little brighter, or even not as loud but still have punch.
Long story short.. it doesn't have to be with a floor pedal like a pod go, helix, kemper or whatever.. you can purchase or maybe even find some freeware VST to get similar results in a DAW, maybe not as user friendly as the Pod has been to me.. but I'd start with an amp/cab sim combination.. From there you can start thinking about compression, EQ VST, or other effects.
Using VST you could record that dry track you're unahppy with and try all sorts of cab sims during playback. You could also use a re-amp box to do the same thing through an effects pedal or amp though.