You don't necessarily need a new base. Just shim the rear ring with a piece or two of aluminum beer can. The CZ uses proprietary rings and integral mounts, so you will be limited in aftermarket long-rnage rings unless you drill and tap, but there are numerous alternatives, with Weaver and Leupold among them
High-power silhouette is shot offhand at 500 yards for the rams. The 6.5 x 55 or .260 are the calibers of choice right now although the 6.5 x 47 is coming on very strong in the new builds.
If you want something significantly better than the CZ you will probably have to go custom or somethng that costs almost as much as a custom. Some of the guys get by with a SAKO in .260 or 6.5 x 55 but it will run you $1500 and only be marginally better than your CZ. The nut cases like me have a Rem 700 action in a Pharr stock with a Hart or similar barrel, Jewell trigger and Metalhead bottom metal, shroud and release. The F-class guys have something similar except with a McMillan stock. The rifle will run $2000-$3000 and you need a good scope - something in the $500-$1000 range like a Leupold FX-3 in 36x. By the way, my silhouette gun is in 6.5 x 55. The F-class guys are leaning toward a 6.5 x 284 which is hotter and will have a shorter barrel life by at least half.
Anyway, shim your rings, put on a good scope and see how you do. Around here the hard part is finding a 1000 yard range. I have to drive 3 to 5 hours to get to one.