Yes, Iyengars are an ofshoot of the Iyers. But Iyer didn't originally mean Smartha Brahmana. Iyer originally just denoted people of the highest jati (subcaste) of the Brahmana varna (caste). Then after the time of Adi Shankaracharya the Smartha philosophy became popular among Iyers. (Smarthas technically worship five gods whom they believe are manifestations of Ishvara, but in practical terms they're often called Shaivite, both because Ishvara is usually considered to be synonymous with Shiva and because Shiva is the god that they typically worship more than the other four. Also note that Smartha originally meant "follower of Smriti", and in that sense pretty much all Hindus in the Kali Yuga are Smartha, but I'm using it in the modern sense of the Smartha philosophy of Adi Shankaracharya.)
And then after the time of Ramanujacharya, Iyers who converted to his Sri Vaishnava philosophy started calling themselves Iyengar. And so since pretty much all the Vaishnava Iyers (other than non-Visitadvaita Vaishnavas) were calling themselves Iyengars, the ones that still called themselves Iyers were the Smartha Iyers, so the term "Iyer" eventually became synonymous with "Smartha Brahmanas of the highest subcaste", which is the current meaning of the word.