Phillips's opening sentences were: "When the demand for a commodity or service is high relatively to the supply of it we expect the price to rise . . . . Conversely when the demand is low relatively to the supply we expect the price to fall . . . . It seems plausible that this principle should operate as one of the factors determining the rate of change of money wage rates, which are the price of labour services." That is, he was talking about about relative prices, not the price level. If the "commodity or service" is used to produce other goods and services, there will be some spillover into the general price level, but it won't wipe out the change in relative prices. In the case of the "price of labour," a given amount of labor can be exchanged for more goods--real wages will rise.
What difference does it make? Here's a regression of change in the consumer price index (inflation) on change in the consumer price index last year and 1/unemployment (the time span is 1948-2015):
The estimate for 1/unemployment is not significantly different from zero (t=1.3).
Here's a regression of change in wages on change in the CPI last year and 1/unemployment:
The estimate for 1/unemployment is significantly different from zero (t=3.5).
I used change in manufacturing wages because that has been collected for the whole period. A broader measure of change in wages (private, non-supervisory) is available starting in 1964. The results:
The estimated effect of unemployment is almost the same and the t-ratio is actually larger.
Finally, what about the "hence" connecting wage increases and inflation?
So only a fraction of wage increases get passed through to price increases. The primary result of low unemployment is higher real wages, as Phillips suggested. Oddly, this point seems to have been forgotten as more sophisticated models were developed.