The time to prune all shrubs / tree's or woody stem / branches is either late summer / early autumn as the plat begins to slow the sap / moisture from continuing on upwards to feed the growing parts of the plants or the other time is early spring just as the sap in about / or has began to rise and feed the plant the moisture it needs to gain energy to grow up / outwards and produce either leaf or flower if it is that type of plant.
Of-course there is always the exception to the rule with some type of shrub . Tree's/ shrubs like Rhododendrons that actually make the flowering buds needed for the following year, these buds are filling out after the older flowers have died off, therefore you really don't want to remove things like flower buds unless you want to wait a few years for the shrub to recover and make new buds as it re-grows.
Hollies are normally as tough as old boots, so I would wait till about March / April time, and give it a prune then, earlier depending on where you live.
IF you feel you should remove more branches / stems after a few weeks, then re prune end of summer.
Many years ago, old farmers would cut there hedges each spring by removing the one side of the hedge one year and the other side the following year, this was to prevent lots of shock to the shrub / tree's and made sure the plant was not killed off should any harsh weather return before the shrub had time to recover and put on it's spring growth.
Keep in mind Hollies are normally evergreen but they do have a rest period in winter.
When the weather warms up I try to avoid heavy pruning as the wounds cause an escape route for the natural moisture held with-in the plant to weep out and sometimes part of the plant just dies off or never recovers, but like I say, most Hollies are tough and after pruning, you will see new tender growth appear within a month or so.
I think there is a plant search on this site so maybe if you search for the Holly you have, there could be better info for you to consider.
Good luck. WeeNel.
Ilex vomitoria (Yaupon) is one of the more durable shrubs, able to come back from all sorts of pruning.
Still, I would wait until you know the new growth will not get hit by frost.
Since they are not grown for flowers, there is no worry about pruning for that reason.
Pruning small sections, such as 25% this year, then more a couple of months later or even next year is a good way to get used to seeing how the plant responds to pruning without risking anything.
Around here they are most often hedged, so get sheared several times a year, but not much taken off each time. They are not fast growers.
While Yaupon Holly (Ilex vomitoria) is not grown for showy flowers (as with most of the Ilex clan), it IS grown for fruit display - which forms as a result of flowering and subsequent pollination.
I don't grow this plant in KY, but I do grow a lot of other species of holly. I would suggest pruning after the plant starts setting fruit - usually by summer months after spring/late spring flowering period. Then, you can decide where to make pruning cuts for form/direction of growth, and decide how much fruit you want to keep with the plant. Of course, this is dependent on whether you have a male-flowered or female-flowered plant - since plants of Ilex are notoriously dioecious.
You could also wait till fall/winter holiday season, and cut fruiting branches for decorations.
Finally: you could post your question at the Holly Society of America website - http://www.hollysocam.org/ - where expert holly enthusiasts can provide you further information.
Paul Rogers (Roots of Wisdom) advises that damaged/broken limbs should be pruned away immediately, regardless the season. Pruning of woody shrubs can start anytime in late winter/early spring - and if shrubs require 'neatening' again, no later than mid-July (the 15th is the cut-off - gives cuts time to harden off before the plants start to wind down for the winter). Flowering woody shrubs excepted (rhododendren, etc). These must be pruned when necessary immediately upon/after flowering or risk losing flowers the following year.
I have heeded this advice for over 25 years without disappointment. My holly and rhods are shapely and healthy, flower and seed profusely.