Geokobbi, When you ask about the second highest mountain in Mali, I am almost certain that it would be one of the mountains near Hombori. There would not be that much difference in height between Hombori Tondo and the others in the same area. Near (ie within 2-3 km) the village of Hombori, there would be perhaps 4 or 5 other high mountains, and further down the road (about 13 km and not 30km as noted in the Lonely Planet West Africa book) towards Bamako, are a series of 3-4 other mountains which you could say are 'high'. These include The Hand of Fatma. Unfortunately for you, who want to climb/walk up them, they all either require rock climbing ropes and equipment, or are not allowed to be climbed due to the locals animist beliefs. My best guess, is that the highest point in Mali, which you are allowed to climb, and can get to without special equipment, is the slightly lower conical peak next to Hombori Tondo. This is the one which I climbed/walked most (but not all) of the way, late last year. I was in the company of another traveller, and although we felt we could get to the top, we thought it may be a little difficult descending, and being aware that it is a long way to a hospital, we decided not to go to the very top. I have no doubt that you could get to the top, if you were careful, without the need for any special equipment.
As for Mont Bagzane, in Niger, I am almost certain that this is the highest point in Niger. -some maps note that it is 2022 metres, including the recent Times Atlas. I would also be confidant that you could climb/walk to the top of this peak, if you knew which one it was. The peak does not stand out in an obvious manner, but is amongst a number of peaks in the Aire Massif. At the time I was there in January this year, it was (and probably still is) a requirement that you have a guide while in the area north of Agadez -so you would need to make sure that the guide was aware of which peak was actually Mont Bagzane. It is an amazing area covered in volcanic boulders and rocks, and dotted with villages. It can get cold and windy at night depending on the time of year you intend going there.
As for other high peaks, Emi Koussi 3415 metres, in the Tibesti mountains in the north of Chad would be spectacular, as is the whole region. May be one day when it gets a little easier and safer to get there.........