The direct imaging of extrasolar giant planets demands the highest possible contrasts (dH ~10 magnitudes) at the smallest angular separations (~0.1) from the star. We present an adaptive optics observing method, called star-hopping, recently offered as standard queue observing for the SPHERE instrument at the VLT. The method uses reference difference imaging (RDI) but unlike earlier works, obtains images of a reference star for PSF subtraction, within minutes of observing the target star. We aim to significantly gain in contrast over the conventional angular differencing imaging (ADI) method, to search for a fifth planet at separations less than 10 au, interior to the four giant planets of the HR 8799 system. We obtained a total of 4.5 hours of simultaneous integral field spectroscopy (R~30, Y-H band with IFS) and dual-band imaging (K1 and K2-band with IRDIS) of the HR 8799 system and a reference star. The reference star was observed for ~1/3 of the total time, and should have dR~1 mag and separated on sky by ~1-2 deg. The star hops were made every 6-10 minutes, with only 1 minute gaps in on-sky integration per hop. We did not detect the hypothetical fifth planet at the most plausible separations, 7.5 and 9.7 au, down to mass limits of 3.6 MJup high signal-to-noise ratios. As noted in previous works, the planet spectra are matched very closely by some red field dwarfs. We also demonstrated that with star-hopping RDI, the contrast improvement at 0.1 separation can be up to 2 magnitudes. Since ADI, meridian transit and the concomitant sky rotation are not needed, the time of observation can be chosen from within a 2-3 times larger window. In general, star-hopping can be used for stars fainter than R=4 magnitudes, since for these a reference star of suitable brightness and separation is usually available. The reduction software used in this paper has been made available online.
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