I would say, first try investing on the E11 and since you are on budget you can then decide/buy the source(mp3player) as i cant say/comment on your xperia's SQ but all we can say the amp does some justice for sure when connected between the phone and ear/headphones. Having said that,you can first try out with your phone and hear yourself how it takes.
Though E11 is bit pricey to E06, its a wise futuristic investment actually,that it can handle any IEMs and most of the Over the ear,full size headphones(even 300 ohm power hungry ones). Features eq,replaceable nokia bl-5b battery etc and excellent form factor to match most of the source like ipods,touch,fiio,ibasso etc when piggy backed.
Whats ur overall budget?looks like you can stretch for both,upto $150......is my understanding correct? That way can give sone tips.
Hahahahaha, no man, what I am doing right now is stretch my budget, and then some... I think my phone's SQ is good, but not great, because sometimes when I listen to music with the same files/IEMs on my laptop (it's a business laptop, no fancy audio card here), I find the SQ to be better, so there is definitely some room for improvement. This is what got me thinking about changing/modifying my source in the first place. I'v found a place nearby where they keep some FiiO amps, but they did not agree to let me sample them before purchasing. I'll push them a bit more for it, let's see.
This raises another question for me, considering I'll be using an amp, with a decent IEM like a Vsonic VSD3S, won't it accentuate the distortion inherent in compressed mp3 Files?