In terms of services trade, it will not be easy for the UK to get better terms in a future UK-South Korea FTA than the current level of liberalisation. For example, in chapter 7 of the EU-South Korea FTA, on Trade in Services, Establishment and Electronic Commerce, each party is required to extend to the other any market liberalisation that it may grant in any future FTA that it signs with another third country.
This implies that South Korea cannot agree any further services market access to the UK, in a future UK-South Korea agreement, without also extending it to the EU.
In addition, if South Korea extends market access to the UK, the US may demand a similar level of access under its agreement with Korea. Ever since Korea negotiated the respective FTAs with the EU and the US in parallel, the US has tried to make sure that the US obtain a similar level of market access to Korea to that granted to the EU.
As a result, it is unlikely that South Korea would be willing to offer better market access to the UK than that currently granted under the EU-South Korea FTA.