Technically, you could do System Conversion (lift & shift) or New Implementation (greenfield implementation). As you already said, you're looking more into a fresh install, please see the link I posted above.
Why starting fresh?
Although it's correct that you probably don't want to take all the history with you, this is a big chance! The chance to start fresh. With standard SAP S/4HANA processes, with clean and valid data,
However, please use the links below to find more information on both scenarios including recorded learning sessions.
Overview Transition to SAP S/4HANA:
S/4HANA Cookbook:
http://scn.sap.com/docs/DOC-64980
New Implementation:
Download the S/4HANA On-premise Best Practices package in the SAP Service Marketplace:
http://service.sap.com/public/rds-dm2s4op
Here is more information regarding Rapid Data Migration to S/4HANA OP which is used for New Implementations. The recording of my ES Academy session is available under this link:
https://service.sap.com/sap/bc/bsp/spn/esa_redirect/index.htm?gotocourse=X&courseid=70306481
Please also check out the below blog posts to get more information about the New Implementation:
System Conversion:
http://scn.sap.com/docs/DOC-68976
More to read:
http://scn.sap.com/community/dm/blog/2015/08/05/introducing-the-data-migration-guys
http://scn.sap.com/community/dm
https://www.linkedin.com/pulse/sap-data-migration-from-lsmw-cockpit-frank-densborn