Several intermediate states of the reaction channels e(+)e(-) -> pi(+)pi(-) D-0(D) over bar (0) and e(+)e(-) -> pi(+)pi-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 psi(3770) signal is seen in the DD invariant mass spectrum, with a statistical significance of 5.2 sigma at root s = 4.42 GeV. There is also evidence for this resonance at root s = 4.26 and 4.36 GeV with statistical significance of 3.2 sigma and 3.3 sigma, respectively. In addition, the Born cross section of e(+)e(-) ->pi(+)pi(-)psi(3770) is measured. The proposed heavy-quark-spin-symmetry partner of the X(3872), the state X-2(4013), is also searched for in the D (D) over bar invariant mass spectra. No obvious signal is found. The upper limit of the Born cross section of the process e(+)e(-) -> rho X-0(2)(4013) combined with the branching fraction is measured. Also, the processes e(+)e(-) -> D-1(2420)(D) over bar c.c. are investigated. The neutral mode with D-1(2420)(0) -> D-0 pi(+)pi(-) is reported with statistical significance of 7.4 sigma at root s = 4.42 GeV for the first time, and evidence with statistical significance of 3.2 sigma and 3.3 sigma at root s = 4.36 and 4.60 GeV is seen, respectively. No evident signal for the process e(+)e(-) -> D-1(2420)(0)(D) over bar (0) + c.c., D-1(2420)(0) -> D*(+)pi(-) is reported. Evidence for e(+)e(-) -> D-1(2420)+D- + c.c., D-1(2420)(+) -> D+pi(+)pi(-) is reported with statistical significance of 3.1 sigma and 3.0 sigma at root s. = 4.36 and 4.42 GeV, respectively.