How to estimate all percentile values (or distribution) from only decile values?

Background: I have two datasets with income estimates, one for the GLOBAL population, one for a LOCAL population (LOCAL only a small fraction of GLOBAL, on the order of 1/100).

Datapoints: I have only certain percentile values, mostly deciles (10th, 20th, 25th, 30th, 40th, 50th, 60th, 70th, 75th, 80th, 90th*). I also have the mean for both populations. [*90th not for LOCAL]

Goal: I'm interested in learning what fraction of the LOCAL population falls into each GLOBAL "decile band", so to speak, i.e. what fraction of the LOCAL population falls in between each of the NATIONAL percentile values that I do have.

Suggested trivial strategy: draw a straight line between each of the LOCAL percentile datapoints that I do have, and interpolate what the LOCAL 11th percentile is, and the 12th, and the 13th, and so on. Matching up to GLOBAL cutoffs will be trivial.

Additional info: More data might be available in past years estimates, or in more detailed breakdowns (income by gender, by employment status, + the proportion of population in each category, e.g. percent male etc.). I cannot access the raw data, only summary statistics such as this.

QUESTION: Is there a strategy likely to yield more accurate results than my suggested trivial strategy above? Is it worth trying to build a distribution of some sort rather than drawing straight lines? If so, how to know which one?

Thanks!