I have found that patches that work well with a slotted tip get stuck or are hard to push with a jag.
I use
these patches with a brass jag and a Tipton carbon rod.
These patches are about like cheap flannel bed sheet thickness and are 1" square.
With a slotted tip I could use 1 1/4" square patches.
Both bags of patches are marked .22cal/.223.
I have started to use bore snakes too. They work pretty good and are fast. I will not always patch until spotless. I run the bore snake through 3-4 times and on the last trip through I put some oil on the end of the snake. I usually dip the snake into the Hoppe's to get about 1" wet just behind the brush section.
I bought my bore snakes on eBay. I have 2 of each size I need. They work excellent and were a fraction of the cost of the Hoppe's ones. The eBay ones are the same color as the Hoppe's ones which makes me think they were made in the same sweatshop in China.
I bought one the the Army issue ODIS kits. Honestly it kind of sucks. Good enough for emergency or field use but not for bench cleaning. It sits in a bug out bag.