3 months is the recommended time in therapy, but it depends on when your therapist thinks (or knows you well enough to say) that you are ready. Some of them will adhere strictly to the SOC, some will take them as guidelines.
As for factors that may come into it - it will depend on you, your history, your age, if there are any other issues that need to be seen to (other than GID) and how far along you are in other (non-medical) parts of transition. A good therapist will look at you and your situation as a whole, and take it from there.
Trend wise, I've noticed the people who got T in a short-ish timeframe were (when starting therapy) :
- Older (mid 20's or older)
- Already living as male full time (with legal name change) for some time (6+ months)
- Had no other issues (social phobia, depression, etc) or found living as male fixed those issues
- Were out to their families, and male at work / place of study
- Were financially independant with their own income (or near enough to it - ie, uni or college students)
- Holding down full time work (or full time study)
- Identified as male since childhood
- Were pretty damn binary conforming
- Went in with absolutely no confusion about what they were doing, and had already done their homework regarding the medical effects (and side effects) of T
- Are completely honest about their past, even if they tried living as female, and even if it's something they think might "work against them".
It may not be "fair" or scientific, but I also think the impression or "vibe" the therapist gets off you counts for more than they let on, too.
Again, that's just a general trend I've noticed. Everybody is different, and some people really benefit from being in therapy longer than others. It's better for you in the long run to find a good, experianced therapist that you click with than it is to find someone that will just write a letter quickly.