Those who have read the pre-Pacelli texts for Palm Sunday will have noticed that olive branches feature just as much as palm branches. This is hardly surprising. The Lord rode in from 'Mount Olivet'; in the Greek to oros ton elaion (S Augustine cheekily proposed translating this as 'the Mount of Chrism'). So when we are told that the crowd cut (ekopton) branches from the trees, we might well conclude that olive branches are largely meant.
These texts allude to the Olive sprig which Noe (Noah) received from his enterprising dove; and explain to us that "surculi olivarum spiritualem unctionem advenisse quodammodo clamant" -- the twigs cry out that a spritual anointing has arrived.
Indeed it has arrived.
Because: this symbolism of the Olive makes a link with the Consecration of the Chrism, traditionally associated with Maundy Thursday. That is when the three oils are blessed or consecrated by the Bishop, who used to concelebrate this Consecration with twelve chasubled presbyters. In the good old truly collegial days before the Council, all twelve of these priests breathed the Spirit, which they collegially share with their Pontiff, upon the Oil. (I suppose the innovators excluded such edifying traditions because they had to find room in the Rite for their own new idea of having all the prebyterium renewing its 'ordination vows'.)
This Chrism will be used in the rites of Christian initiation at the Easter Vigil, to confer the Sacrament of Confirmation (or Consignation). This use is probably to be deemed Apostolic, since, in the Greco-Roman world, the use of oils regularly accompanied Washings.
In the Latin Church, Oil is also used in Ordinations ... and in some other solemn consecrations. In the Church of England, the Monarch is still anointed, although whether the oil is regarded as Chrism, I am unsure (I believe it is the prerogative of the Dean of Westminster to 'bless' it). Now that Chrism is consecrated in (I think) every Church of England diocese, I wonder what the highly secret File marked "Next Coronation" prescribes.
Our last Monarch both de iure and de facto, King James VII and II, was, I think, Crowned by the Nuncio according to the Roman Pontifical; and then crowned again by the Archbishop of Canterbury!
In the Church of England, once they stop prohibiting things, they do tend to go to the other extreme and become daftly fanatical about them, poor perplexing poppets. So when I was to be licensed as pp of S Thomas the Martyr in Oxford, my last pre-Ordinariate ministry, the Archdeacon asked me if I would like to be anointed!! "Good Heavens, No!!" I cried. "Let's keep it all as simple as possible".
Traditionally, the Oil of Chrism has been seen as so sacred that S Cyril of Jerusalem (whom Gregory Dix once described as a 'very extreme' churchman) regarded its consecration as being analogous to the transsubstantiation of the Eucharistic Elements, with the result, in his view, that it 'became' (ginomenon) the charisma of Christ and 'effective of His Godhead' (tes autou theotetos energetikon). (When this Catechesis was incorporated into the post-Conciliar Liturgia Horarum, that exotic paragraph was eliminated!!)
Among some Byzantines, the confection of the Chrism is restricted to Patriarch. One can see why.