There is no 'CFI rule' @ any airline (or anywhere else, really). If you can build time w/o getting a CFI and your only motivation to get a CFI is to satisfy a perceived rule, then don't get one.
There is most certainly a 'multi' rule @ airlines (and most other professional flying jobs), though. Professionals fly multi engine aircraft, generally, so employers hire those who have that experience. Each hour of multi is much more valuable than an hour of SEL. How much more valuable depends on the person, the day, the mood of the person interviewing you, etc - there is no 'rule', other than the more of of what they do that you've already done, the better (at the entry level professional jobs).
The bigger question is the marketplace. If you don't have your multi now, you need to think about what you want out of flying. Virtually nobody is hiring right now due to the economy and age 65 legislation. Hiring for pilots such as yourself will likely start up again in 1-2 years. @ that point, there will be a good amount of pilots who've been struggling as CFI's for the past 2-3 years. You need to have a better resume then them, b/c you likely won't have as many hours as them. There may be new FAA requirements to have more total time to fly the right seat of an airliner too - likely 750 TT if the FAA's view prevails, or 1500 TT and an ATP if Congress and the pilot unions prevail.
No matter what you do, it's a gamble. Each hour of experience you get is valuable, and CFI'ing can make you a better pilot, as can doing other things.
Your WSO time and masters will be the HUGE difference b/t you and the rest of the guys you're competing against. You have flying, leadership, and decision-making experience that is unparalleled in your marketplace (entry level, regional FO). Don't be afraid to talk about being a WSO @ every interview you go to.
Personally, if you don't have time building issues, I don't think you need to get your CFI, but if you WANT to, have at it. Your WSO and Military officer experience is, to me, more valuable than CFI experience for showing you have leadership and decision making skills. The flying skills though, the CFI could help you there if you don't have them already.
There are no 'rules'. Good luck in your quest, and always put safety first.
Last edited by Sniper; 02-13-2010 at 08:24 AM.
Reason: added WSO comment