I was at St. Demetrios in Knoxville. Not sure which body they are affilliated with, just read the preface of their Qurbana book and recognized the Liturgy as Syrian (which the book noted). Several of the clergy were wearing phiros, which they removed for the Gospel and the Lord's Prayer, but the principal celebrant wore something that looked a touch more like a short russian hat (can't recall the spelling of the name). I have more personal experience with the Antiochean Maronite Rite which makes use of a very tight-fitting monastic hood (as opposed to a zuchetto or phiro) and the Mitre (for Chorbishops and Bishops). They remove both the hood and mitre for the Anaphora, but wear them in all other prayers.
Upon some investigation, it seems that that church is not affiliated with either the Church of Antioch or the Church in India, both of which follow the Syriac rite, nor with any of the Catholic Syriac rite Churches. I could not personally call it a "Syrian" church, and I'm not surprised there are divergent practices there.
I cannot speak for Maronite tradition: I am not familiar with them, and they are a slightly different breed of Syrian.
Bishops in the Syrian tradition are interesting, as they have more than one "headcovering". They use the phiro, the monastic hood (eskimo), and a vestment called a masnaphto, which corresponds to the Greek mitre, but is more like a hood. During certain parts of the Liturgy, only the latter is ever removed. The others stay on.