Since I already pray the Liturgy of the Hours daily, its very format automatically sets it apart for the Easter Octave.
Agreed. I mainly pray from the traditional Monastic Office these days, though I have spent a lot of time with the Liturgy of the Hours as well.
Somewhat related: I think one of my favorite things about the EF and traditional Monastic offices is that they retain the Octave of Pentecost, so that the Easter Season ends as Trinity Sunday begins. After Advent, Christmas, Epiphany, (Septuagesima,) Lent, Passiontide, the Triduum, Easter, Ascension, and finally the Holy Spirit’s descent at Pentecost, it feels like those last few days of the Paschal season are like a final
hurrah culminating with the change of seasons at Trinity Sunday when we celebrate the most central tenet of our faith. Plus I think the birth of the Church merits a good octave

Even when I’m praying the LOTH, I usually switch to an older office for the week following Pentecost for this reason. I have heard of some priests saying votive Masses of the Holy Spirit during that week to retain an “octave feeling,” though I’ve never seen it done myself.
This year I prayed the Sorrowful Mysteries of the Rosary daily throughout Passiontide and the Triduum (2 weeks), so for the Easter Octave (as well as Ascension Thursday through Trinity Sunday) I plan to pray the Glorious Mysteries exclusively to really immerse myself in the mystery of the Resurrection.
I’m also praying the Divine Mercy novena, which happens to fall during the Easter Octave. I don’t pray the DM Chaplet daily anymore but I like to pray it throughout this week along with the novena prayers.