With some exceptions, Japan Rail base fare tickets allow stopovers. There is an excellent answer that explains when a ticket is eligible for stopovers, but it is not clear from either that answer or the linked JR website whether skipping forward is allowed during a stopover.
Suppose I were to take the train from Nagasaki to Fukuoka and stop to visit the Yoshinogari Historical Park on my way. The park is located between two train stations on the line that connects Nagasaki and Fukuoka, so my most efficient route would be to board the train at Nagasaki, alight at Kanzaki (the station closer to Nagasaki), tour the park, board another train at Yoshinogarikoen (the station closer to Fukuoka) and proceed to Hakata (or another station within the Fukuoka city zone).
Can I do this with a single base fare ticket from Nagasaki to Hakata? Or do I have to return to the same station where I exited the ticket gates to resume my journey?