CIC is more than just basic training - it includes specific Inf training. A TA Inf soldier would not expect to transfer to the R Sigs without doing the special to arm course, so why should it be any different the other way around?
You would need to do all the inf recruiting stuff again. CMSR, in this callsigns opinion, is an overveiw of basic soldiering. CIC is more focused on all things inf (sec attacks, bcd, ranges, etc). I thought the same when I transferred from a corp to inf but found the recruit build up to be more involved than my CMSR was. There where a couple of us in the same situation and we all agreed on that.
Currently CIC(TA) follows on from TSC(A) + TSC(Inf). The first question is how long ago did you do CMS(R) as there have been many changes. TSC(Inf) will take you up to LF9 shooting and a higher standard of Phys. If you completed Phase 1 C then you will have covered all the BCDs, CBRN, Mapreading etc but you may not have covered the field craft in enough detail. My advice is that you contact your RTC and get yourself on a TSC(Inf) to bring you up to speed.
Remember, CIC(TA) is about 25% Phase 1 and 75% Phase 2 training. Good luck.......