IMHO, you should create a custom field that is only tracked against sub-tasks to track "hours remaining" specifically. Be careful about going overboard and tracking vs actuals as that can lead to some seriously weaponized metrics. In general, this sort of data is a gold mine for micro-managers; be very careful not to give such managers ammunition to harass an otherwise healthy team :).
You can refer to this guide:
https://confluence.atlassian.com/display/JIRA052/Adding+a+Custom+Field
I would suggest you create the field as a float, but try not to get hung up estimating tasks too granularly :).
You likely have to reindex, but you can do that in the background to keep from interfering with others' work.
Beyond adding the custom field, I would suggest configuring your Agile board card format to show the hours remaining. You'll have to go few a couple hoops to make sure the new field is available in the custom fields group, but Jira prompts you through that process.
At the end of this, you'll have enabled the developers to break a user story into sub-tasks and put hour estimates on those sub-tasks without affecting the story level estimate.
Last, you may be working in a domain that is well suited for temporal estimates. Plenty of very simple, repetitive work lends itself to being easily and precisely estimated in this way; however, if your project is like the majority out their, it's not so simple. Consider reading up and choosing relative estimation techniques to ensure you're getting enough accuracy in your estimates without incurring the overhead and error of forming precise estimates.
