I don't think Fulmer orchestrated anything pre-Currie firing. Not because he did not want the AD job that ultimately went to Currie (he absolutely did) but because I don't think he had the leverage to pull that off at the time.
Once Currie was fired, Fulmer and his booster allies had leverage at that point and he absolutely got involved, which ultimately culminated in him getting the AD job.
Thornton was in touch with Currie's bosses telling them Fulmer was ready to roll. He was also sabotaging Currie's attempt to hire others and went full nuclear and pulled the coup when Currie hired Leach. It wasn't Phil's first sabotaging trip around the back rooms while at UT.