There are lattices where your requirement cannot be met. In fact if you fix any lattice $\Lambda_0$ and you dilate it by a factor of $R$, then its determinant gets scaled by $R^n$, while its shortest lattice vector gets scaled by $R$. However, your inequality $\|v_1\|\ll_n\det(\Lambda)^{1/(n+1)}$ would yield that the shortest lattice length grows at most at the rate of $R^{n/(n+1)}$, which is a contradiction.

You can get further lower bounds on the lengths of the basis vectors by using Minkowski's theorem that the successive minima of $\Lambda$ satisfy
$$\lambda_1\lambda_2\dots\lambda_n\asymp_n\det(\Lambda).$$
Note also that the reduced basis produced by the LLL algorithm is optimal in the sense that the length of the $i$-th basis vector is $\asymp_n\lambda_i$. This is explicitly mentioned in the original paper of Lenstra-Lenstra-Lovász (1982), see their remark below (1.13).