The HG612 has a pretty old DSL chipset.
My chipset of choice is the BCM63168
It syncs higher than any other chipset I've tried.
I'd confidently say it's preferred chipset for the majority on this forum.
Kitz has compiled a list with most of the modem/routers using Broadcom DSL chipsets.
https://wiki.kitz.co.uk/index.php/BCM_routersI find the Zyxel VMG1312-B10
A (Not the VMG1312-B10
D) syncs the highest on my line.
It's no longer manufactured so is becoming harder to source at a reasonable price. You can pick them up on EBay for about £40 new.
The Billion 8800NL R1 (NOT the newer R2) is also a highly recommended modem on these forums. It also has the BCM63168 chipset.
There's a newer BCM chipset, the BCM631
38 that seems to perform as good as the 63168 for some users.
I found it synced slightly lower than my Zyxel though.
The choice of modems with this newer chipset is still quite limited.
After saying all that, you may find you see no improvement what so ever compared to the HH6/smart-hub.
I've never tried it myself (sold the 1 BT gave me, still sealed) but it seems to sync pretty well.