Safe generation of $k$ points on a curve such that the mutual discrete logs are hard? I have a multiplicative group $G$ of prime order $p$ implemented using a twisted Edwards curve (similar to Ed25519). 

I want to compute a set of $k$ distinct points $P_1,...,P_k$ that generate $G$, such that the mutual discrete logarithms of these points are hard to compute. I.e., for any distinct $P_i,P_j\in \{P_1,...,P_k\}$ it should be hard to find $s$ such that $P_i=s\cdot P_j$. 

To obtain such a point $P_i$, I do the following steps:


I select $y$,
I use $y$ to compute $x$,
I verify if $(x,y)$ is on the curve,
I verify if the order of $(x,y)$ is $p$.


If $(x,y)$ is not valid, I retry with a different $y$. I repeat this process until I have found my $k$ points.

My question: 

Can I generate the points by simply increasing $y$ (e.g., $y=3,4,5,...$) or will this affect the security if the $y$ coordinates of all my points are in a very small range?
It would be more confidence-inspiring to choose a hash-to-curve method and pass consecutive integers through that.  There is a current (2019-05-16) internet-draft surveying hash-to-curve techniques with references, draft-irtf-cfrg-hash-to-curve.  The most popular method for Ed25519 is Elligator 2.  Since invertibility is not required for your application, you could also just use the hash-and-pray technique: for each $i$, find the smallest $j$ such that $H(i) + j$ is an $x$ coordinate on the curve, where $H(x)$ is (say) $\operatorname{SHAKE128-512}(x) \bmod{2^{255} - 19}$.