I am reading a paper "A Simple Way to Estimate Bid-Ask Spreads from Daily High and Low Prices" cf.A Simple Way to Estimate Bid-Ask Spreads from Daily High and Low Prices

The authors proposed the method of estimation the bid-ask spread from high and low prices of consecutive two days.

From what I can understand, there is an important assumption there that the prices follow geometric Brownian motion and, therefore, the true variance over a 2-day period is twice as large as the expectation of the variance over a single day. This property is used for the spread estimation.

Next, assume that I have more data, than just high and low prices, say, 10 min bars.

Will it improve the spread estimator if I use high and low prices of consecutive two 10 min bars instead on days? Does it contradict to the derivation for daily case?

If you have access to intraday data, they are better ways to estimate the bid-ask spread. If you have Open, High, Low and Close price on each 5min bin $$b$$ (or any other interval): the Close of the previous bin and the Open of this one are consecutive. Hence $$dP(b)=C(b-1)-O(b)$$ allows to define an estimate $$\psi(b)$$ of the bid-ask spread $$\psi(b):=\min_{b:\, |dP(b)|>0} |dP(b)|.$$
It is not defined on every bin of each day (sometimes $$dP(b)=0$$), but often you have several of them. You can average them to obtain an estimate for the bid-ask of the day.
Of course you can add an estimate deduced from High and Low of the bin, but it is clearly worst than this $$\phi$$.