Particles of fast solar wind in the vicinity of the heliospheric current sheet (HCS) or in a front of interplanetary coronal mass ejections (ICMEs) often reveal very peculiar energy or velocity profiles, density distributions with double or triple peaks, and well-defined streams of electrons occurring around or far away from these events. In order to interpret the parameters of energetic particles (both ions and electrons) measured by the WIND spacecraft during the HCS crossings, a comparison of the data was carried out with 3-D particle-in-cell (PIC) simulations for the relevant magnetic topology (Zharkova and Khabarova, 2012). The simulations showed that all the observed particle-energy distributions, densities, ion peak velocities, electron pitch angles and directivities can be fitted with the same model if the heliospheric current sheet is in a status of continuous magnetic reconnection. In this paper we present further observations of the solar-wind particles being accelerated to rather higher energies while passing through the HCS and the evidence that this acceleration happens well before the appearance of the corotating interacting region (CIR), which passes through the spacecraft position hours later. We show that the measured particle characteristics (ion velocity, electron pitch angles and the distance at which electrons are turned from the HCS) are in agreement with the simulations of additional particle acceleration in a reconnecting HCS with a strong guiding field as measured by WIND. A few examples are also presented showing additional acceleration of solar-wind particles during their passage through current sheets formed in a front of ICMEs. This additional acceleration at the ICME current sheets can explain the anticorrelation of ion and electron fluxes frequently observed around the ICME's leading front. Furthermore, it may provide a plausible explanation of the appearance of bidirectional "strahls" (field-aligned most energetic suprathermal electrons) at the leading edge of ICMEs as energetic electrons generated during a magnetic reconnection at the ICME-front current sheet.