How hard is it to obtain a TS (SCI) clearance if you currently hold a secret?

It's not likely to speed up the process although you'll have a lot of your references in there. I was put down as a reference for a soldier in my Reserve unit they were upgrading the clearance on and I'm pretty sure it still took the investigators about 8 months to get to me for an interview.

Your S2 or security office type people will usually take care of the initiation of the paperwork. It will be best on your part to start getting together all the people you need references from for the past 10 years for a TS investigation and filling out a copy of an SF-86 on paper. This way, when you need to fill in the electronic version, you will have it all ready to go as the window to enter the information once it opens is usually only a week and you don't want to be scrambling to get current information for everyone in order for them to be contacted at the last minute.