How can I reach the two minima connected by a first-order saddle point (transition state) using the Gaussian Software
how can I use the EF option of Gaussian to perform an optimization along an imaginary mode starting from a transition structure?


I have never used the EF mode of Gaussian 09, but...


  Is there a more elegant way, circumventing manual structural modifications, to yield the two minima connected by the TS?


I believe the standard way of proving that your TS is the TS you want is by an intrinsic reaction coordinate calculation:

http://gaussian.com/irc/

Make sure to check out the "options" tab. A decent write-up on a small system is here:

http://www.cup.uni-muenchen.de/ch/compchem/geom/irc5.html

Basically, you need to restart the calculation using the checkpoint (chk) file of your TS calculation, and as you calculated the vibrational spectrum, you can use "irc=rcfc" to read the force constants from the restart file. By default the IRC calculation does 10 points in both directions of the "reaction coordinate" starting from the TS, you can change this using MaxPoints=n, and you can also specify which direction you want to study using other keywords listed in the "options" tab on the Gaussian website.

Opening the logfile in Gaussview, you can see the IRC profile - the same way you can look at the optimization profile of an "opt" calculation. I'd follow up on the IRC by optimizing each of the end-points of the profile using a standard "opt" calculation, just to make sure.

Hope that helps.