Disclosed are a drilling rig for drilling with a following pipe pile and a construction method for the following pipe pile. The drilling rig is provided with a chassis, a platform, a cab, a power system, a hydraulic system, a control system, upright columns, supporting rods, a vertical winch, a power head, a soil discharge box, a spiral drill stem assembly, a reamer bit, a vibration exciter, an adjusting cylinder, a manned hanging basket, a hanging basket winch, a pressurizing winch, an auxiliary winch and a pile controller, wherein the platform, the cab, the power system, the hydraulic system and the control system are located on the chassis, the upright columns and the supporting rods are mounted on the platform, and the auxiliary winch is used for hoisting heavy objects. The construction method for the following pipe pile comprises following steps: leveling the pile controller after the pile controller is aligned with a pile location and adjusting verticality of upright columns of the drilling rig; mounting a prefabricated pipe pile and a drill and leading the reamer bit to penetrate out of the lower end surface of the prefabricated pipe pile; forwards rotating the power head, leading a reamer wing of the reamer bit to be opened to begin to realize reamer drilling and leading the prefabricated pipe pile to be jacked in afterwards; backwards rotating the power head after a preset depth is achieved, withdrawing the reamer wing of the reamer bit to be closed, lifting out the reamer wing of the reamer bit from the inside of the prefabricated pipe pile and starting the vibration exciter to impact the prefabricated pipe pile; and realizing grouting piling for the sides and the bottom of the pile.