Several intermediate states of the reaction channels e+e-→π+π-D0D̄0 and e+e-→π+π-D+D- are studied using the data samples collected with the BESIII detector at center-of-mass energies above 4.08 GeV. For the first time in this final state, a ψ(3770) signal is seen in the DD̄ invariant mass spectrum, with a statistical significance of 5.2σ at s=4.42 GeV. There is also evidence for this resonance at s=4.26 and 4.36 GeV with statistical significance of 3.2σ and 3.3σ, respectively. In addition, the Born cross section of e+e-→π+π-ψ(3770) is measured. The proposed heavy-quark-spin-symmetry partner of the X(3872), the state X2(4013), is also searched for in the DD̄ invariant mass spectra. No obvious signal is found. The upper limit of the Born cross section of the process e+e-→ρ0X2(4013) combined with the branching fraction is measured. Also, the processes e+e-→D1(2420)D̄+c.c. are investigated. The neutral mode with D1(2420)0→D0π+π- is reported with statistical significance of 7.4σ at s=4.42 GeV for the first time, and evidence with statistical significance of 3.2σ and 3.3σ at s=4.36 and 4.60 GeV is seen, respectively. No evident signal for the process e+e-→D1(2420)0D̄0+c.c.,D1(2420)0→D∗+π- is reported. Evidence for e+e-→D1(2420)+D-+c.c.,D1(2420)+→D+π+π- is reported with statistical significance of 3.1σ and 3.0σ at s=4.36 and 4.42 GeV, respectively.