Remove shingles and other roofing material in the area for your new hip roof, down to the basic decking. Set one truss for the new roof in place, hold it plumb, then snap chalk lines on either side from the truss to the roof. These will be the valleys for your hip roof addition. Nail 2-by-6 valley boards along these lines. The length of the added hip roof will determine how many trusses you need before starting your hip framing. Set at least one truss and "jack" trusses -- short trusses cut to fit between the existing roof and new hip roof -- and nail them in place with a ridge board holding them at the top.
Cut 2-by-6 or 2-by-8 ridge boards to run from the truss down to the eaves of the new hip roof addition on either side and in the center. Then cut and nail in 2-by-6 common rafters on 16-inch centers, from the ridge board to the side wall caps of the addition; cut notches in the bottom ends to set the rafter on top of the wall cap and let the rafter overhang the wall.You can avoid cutting by using metal hangers, which nail to the wall cap and hold the rafter at the proper angle.
Install 2-by-6 common rafters on 16-inch centers from the peak to the end of the added hip roof; notch these or use metal hangers to get the same overhang as the edges. Install "hip" rafters at the corners, cut lower than common rafters and typically set on a 45-degree angle from the corner to brace the corner.
Install 2-by-6 joists across the added hip roof. Nail a center board from the old roof to the new outer edge, then nail joists from the center to the sides; use metal joist hangers to hold them. Cut these joists to extend out to the edge of the rafter overhang, fitting beside the rafter. At the end wall, cut and install short joists to fit from the wall cap to the rafter ends. Then add soffit boards to cover the underside of the overhang and finish with fascia or trim boards nailed to rafter ends around the edges of the new hip roof.
Deck the new roof with oriented strand board (OSB); you will have to cut 4-by-8 sheets to fit edges. Nail this to the rafters. Nail metal flashing on the valleys where the new roof meets the old and metal drip edge on all roof edges, then cover the roof with roofing paper, overlapping at the seams. Then shingle the new roof, bending shingles at the valley to cover it and tie old and new shingles together.