Should the Gayatri Mantra be chanted only during Sandhyavandanam? Can it be chanted at any time (perhaps after chanting the primary Acharya’s Taniyan)?  

We perform Gayatri Japa during the time of Sandhyavandanam. Aside from that, the Japa can be done while performing certain Prayaschitta (atonement) rituals. It is not customary to chant it at other times. There is a specific Sankalpam (formal vow of intent) for Gayatri Japa. If it is Sandhyavandanam, the Sankalpam will specify “Pratha Sandhyavandanam” (Morning), etc. There is no benefit in chanting it without knowing the proper Sankalpams. However, one may chant the Japa at other times.