Communion is actually being received *from the altar* (i.e., Christ) which is sometimes called the "Eucharistic Table" though not meant in the Novus Ordo sense. In Catholic England, the altar was commonly referred to as the "Holy Board" (i.e., from the term "room and board" or food). Of course this symbolic language coincides perfectly with other terms such as "Angelic Banquet", "Divine Supper", "Bread of Angels", etc. The Eastern Rites in particular also have a variety of beautifully symbolic terms to describe this "meal" aspect of Holy Communion (which is a fruit of the Holy Sacrifice). Of course we also have reference to Passover Pasch, or eating of the Lamb in Agnus Dei. Such rich realities!
Also, the Communion rail is actually an extension of the altar (thus sometimes called the "altar rail"), and the prescribed Communion cloth an extension of the altar cloth, and in reality, the corporal.
In larger churches that did not have Communion rails (as frequently seen in major Roman basilicas and European cathedrals - NB: this is not due to any N.O. wreckage - it was always like this), the Communion cloth was held by the acolytes under the chins of the communicants (NB: the Communion plate was not introduced until the 1920's, and per a 1929 clarification of the Sacred Congregation of Rite, was to be used *in conjunction* with the cloth).
One other quick point, the faithful should not bury or put their hands under the cloth (i.e., unnecessarily touch or lift it) as this can only disturb the intended flat surface (which acts as safeguard in case the Host is accidentally dropped) and cause it to sloop (thereby potentially causing a dropped Host to slide off the cloth rather than being caught) as well as dirties the pure white linen. J.B. O'Connell in "The Celebration of Mass: A Study of the Rubrics of the Roman Rite" and others (e.g., "Matters Liturgical") point this little directive out.