This question isn't an exact duplicate of a ton of related questions as linked in a comment on the OP, but it is both highly similar to a ton of related questions about tracking incomplete work especially within JIRA and mostly an X/Y problem because it assumes incorrectly that work is permitted to automatically "roll over" across Sprint boundaries within the Scrum framework.
Short Analysis and Recommendations
Since this question or ones like it are founded on an initial fallacy (e.g. that items from the Product Backlog can or should automatically roll over across Sprint boundaries), the shortest response I can think of is that:
- This isn't a JIRA problem because you shouldn't be doing this in either Scrum or Kanban.
- "Accurate time counting" across Sprints seems like a proxy for consumed story points, so while you may be able to do this within JIRA it's solving the wrong problem.
- The team and the process needs to respect time boxes if you're using a time boxing framework like Scrum.
Given the above, you should treat both the Sprint Goal and all work within the Sprint as binary, e.g. done or not done per the Definition of Done. Anything "not done" is returned to the Product Backlog as potential future work based on:
- The feedback from the Sprint Review and the Sprint Retrospective.
- Changing Product Goals or priorities within the Product Backlog.
- Changes resulting from Backlog Refinement activities.
- What is in scope for Sprint Planning in future Sprints.
Don't Let JIRA Drive Your Process
What you should never do is allow a tool (in this case, JIRA) to drive your process. Tools support a team's optimal (or at least chosen) process, but should never define it for you.
By trying to define success as a JIRA report on effort expended or "progress" towards some abstract definition of completion rather than as a binary outcome, you have inverted the delivery-oriented aspects of agility. In other words: Do you have a potentially-deliverable feature or not? If the answer is no, then "progress made" or "effort expended" are irrelevant metrics.
While one might make an argument for work or effort remaining to delivery a given outcome, these need to be re-assessed in the present rather than based on some historical starting point. That is why user stories shouldn't carry history, and why historical estimates should be discarded in favor of future-facing estimates—assuming that the work in question remains useful or valid at all, which is not a given within any agile framework.