It doesn't require a higher TCR (unless there are impurities in the metal) You are mixing heat flux, ambient temperature and heat capacity
up with TCR
Whilst they directly affect each other in terms of the collision process that occur, the atoms in the wire you are using behave the same so doesn't matter whether you use a big fat peice of wire or a tiny thin peice those atoms for that metal type are still structured in the same way and will still resist the same on average over the heating process (assuming same metal from the same manufacturer)
It's the other properties I mentioned that affect the experience you are seeing that's why cracking up the wattage or preheat functions maybe needed
Don't forget the more mass the longer it takes to cool down as well so maybe only the preheat function needs to be modified
PG/VG ratio Also matters VG has a higher boiling point than PG, wicking and airflow and heat dissipation in the chamber also play a big role in the TC experience
The best article I've read on the introduction to TC is the below I'll paste this into my Q&A thread as well
https://www.ecigssa.co.za/guide-to-fine-tuning-temp-control-vaping.t18206/