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It sounds as if you own the splitters already. As ADTech has suggested, a line amp, AKA distribution amplifier would be plenty to drive the insertion losses of the splitters and cable you've described. The Channel Master CM-3410 and Winegard HDA-100 quickly come to mind.
Presuming the splitters are of decent quality, in theory, there should be no difference between the system performance of an amplifier followed by a discrete splitter, vs. an amplifier with the splitter built in. However, in practice there are things to consider.
As a rule of thumb, it's best to use only one splitter, and to spec. the splitter with the exact number of ports needed. If you follow this rule, you'll avoid excessive losses including those caused by daisy chaining three or more splitters. Then there's the cost to be considered. For example, to serve four sets using 2-way splitters, you would need three 2-way splitters, the first to feed the other two. Then you would need more space and connecting cables... more cost and connections to fail. (If you had only 2-way splitters to work with, be sure to connect them in a pyramid not a tandem chain.)
Another rule of thumb: One amplifier can be too many. Two amplifiers are almost always too many. An amplifier can only take so much signal at it's input before it's driven into overload, which causes distortion and noise to dramatically increase. Unless you have calculated the losses caused by cable and splitters and have matched the amp gain to those losses, you risk overloading the second amplifier.
Your installation does not have losses that would justify a second amplifier... in fact, I would try running a larger antenna for it's passive gain. Which brings up another truism: 'Antenna gain is better than amplifier gain.' Antenna gain can't be overloaded, it does not add noise or distortion and it tends to come with the benefit of increased off bore-sight interference rejection (none of which an amplifier can do).
In an ideal world, all coax would be run to a common location. No run would have any splices mid-span. And all the cable would be tagged. This method may cost a bit more than a daisy-chained installation, but it's the most flexible and far simpler to understand years later. If one home-run cable is damaged, only that run is affected, but failure of the first cable in a daisy-chain affects the entire system.
If I where starting from scratch, I'd go with an Antennas Direct DB4e, outside, in the clear, facing 305° magnetic. I'd point an Antennacraft Y5713 at 125°. Then combine the two antennas using an Antennas Direct UHF / VHF Antenna Combiner. Finally, I would try this with a passive 3-way splitter. I would only add a DA if the need was proven.
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