We have recently detected CO lines in the well-known filaments around NGC 1275, the galaxy at the centre of the Perseus cluster of galaxies. These previous observations, with the HERA multi-beam array at the IRAM 30 m telescope enabled us to make a large map of the CO(2-1) line and to see hints of molecular gas far away from the cluster centre. To confirm the presence of CO emission lines in the outer filaments and to study the CO(2-1)/CO(1-0) line ratio, we observed seven regions of interest again with the 30 m telescope in both CO(1-0) and CO(2-1). The regions we observed were: the eastern filament, the horseshoe, the northern filament and a southern extension, all selected from Ha emission line mapping. Molecular gas is detected in all the observed regions. This result confirms the large extent of the cold molecular gas filaments. We discuss the CO(2-1)/CO(1-0) ratios in the filaments. The eastern filament has optically thick gas, whereas further away, the line ratio increases close to values expected for a warmer optically thin medium. We also show CO(1-0) and CO(2-1) lines in 9 regions closer to the centre. The kinematics of the CO is studied here in more detail and confirms that it follows the motions of the warm H-2 gas found in the near-infrared. Finally, we searched for dense gas tracers around 3C 84 and claim here the first detection of HCN(3-2).