One reason not to chain is so that you can more easily control one resistance value without having effects on all the other values in your chain.
Since the receiving device most likely measures voltage, the best chain is one where the resistances are chosen to distribute the voltages across the measurement range.
Your unfinished diagram uses three 1k resistors to make a range of 1k 2k 3k. The difference in voltage between 1k and 2k is unlike the difference in voltage between 2k and 3k. Given enough switches, that difference might not be enough to remove ambiguity in the receiving device.
So I would aim for 1k 2k 4k (multiply by two), or 1k 5k 25k (multiply by five). Assembling these ranges becomes more difficult using a chain, because of how resistors are available in an E series. Assembling them not as a chain can be easier; e.g. 1k 4k7 22k.
This goes back to your test results on how different resistances must be to be differentiated by the device. It is all experimentation, since you don’t know the internal design of the device.