High-sensitivity double-gap phenolic resistive plate chambers (RPCs) are studied for the Phase-2 upgrade of the Compact Muon Solenoid (CMS) muon system at high pseudorapidity . Whereas the present CMS RPCs have a gas gap thickness of 2 mm, we propose to use thinner gas gaps, which will improve the performance of these RPCs. To validate this proposal, we constructed double-gap RPCs with two different gap thicknesses of 1.2 and 1.4 mm by using high-pressure laminated plates having a mean resistivity of about 5 1010 -cm. This paper presents test results using cosmic muons and 137Cs gamma rays. The rate capabilities of these thin-gap RPCs measured with the gamma source exceed the maximum rate expected in the new high- endcap RPCs planned for future Phase-2 runs of the Large Hadron Collider (LHC).