The invention relates to a process and a machine for making a tissue paper web (W) in which the tissue paper web W is passed through an extended nip N formed between an extended nip unit (2) and a Yankee drying cylinder (1) and in which the tissue paper web W is carried on a felt (3) through the extended nip N in such a way that, in the extended nip N, the tissue paper web W contacts the outer surface (4) of the Yankee drying cylinder (1). The web W and the felt (3) are led over a suction roll (5) prior to the extended nip N in such a way that the felt (3) contacts the suction roll (5) and the tissue paper web W is separated from the suction roll (5) by the felt (3). The suction roll (5) has a suction zone (6) over which the felt (3) and the tissue paper web W pass together, and a first hood (7) is arranged opposite the suction roll (5) and partially surrounds the suction roll (5). The first hood (7) has an extension around the suction roll such the first hood (7) covers the entire suction zone (6), and moist hot air is fed from the first hood (7) and sucked through the tissue paper web and the felt (3) by the suction roll (5). The tissue paper web W is directly exposed to the first hood (7) such that the moist hot air reaches the tissue paper web W without passing through a fabric before reaching the tissue paper web W. The Yankee drying cylinder (1) is covered by a second hood (8) which is a Yankee hood which has an air heating and distribution system (9) and hot exhaust air from the second hood (8) is fed through a conduit (10) to the first hood (7) and used to supply the first hood (7) with moist hot air. The moist hot air has a temperature in the range of 130° C.-300° C. and a moisture content of 300 g/kg dry air-1000 g/kg dry air at a rate of 90-130 m3/minute per square meter suction zone area. The moist air is then sucked through the tissue paper web W by the suction roll (5) such that moisture condensates on the tissue paper web W and thereby raises the temperature of the tissue paper web W before the tissue paper web W passes through the extended nip N.