To satisfy the various demands of growing devices and services, emerging high-frequency-based technologies promote near-field wireless communications. Therefore, near-field physical layer security has attracted much attention to facilitate the wireless information security against illegitimate eavesdropping. However, highly correlated channels between legitimate transceivers and eavesdroppers of existing multiple-input multiple-output (MIMO) based near-field secure technologies along with the low degrees of freedom significantly limit the enhancement of security results in wireless communications. To significantly increase the secrecy rates of near-field wireless communications, in this paper we propose the double-reconfigurable-intelligent-surface (RIS) assisted orbital angular momentum (OAM) secure scheme, where RISs with few reflecting elements are easily deployed to reconstruct the direct links blocked by obstacles between the legitimate transceivers, mitigate the inter-mode interference caused by the misalignment of legitimate transceivers, and adjust the OAM beams direction to interfere with eavesdroppers. Meanwhile, due to the unique orthogonality among OAM modes, the OAM-based joint index modulation and artificial noise scheme is proposed to weaken the information acquisition by eavesdroppers while increasing the achievable rate with the low cost of legitimate communications. To maximize the secrecy rate of our proposed scheme, we develop the Riemannian manifold conjugate gradient (RMCG)-based alternative optimization (AO) algorithm to jointly optimize the transmit power allocation of OAM modes and phase shifts of double RISs. Numerical results show that our proposed double-RIS-assisted OAM near-field secure scheme outperforms the existing works in terms of the secrecy rate and the eavesdropper's bit error rate.