$\sum \frac{a_n}{\ln a_n}$ converges $\implies \sum \frac{a_n}{\ln (1+n)}$ converges 
Let $0<a_n<1$ be real numbers. Prove that $\sum \frac{a_n}{\ln a_n}$ converges $\implies \sum \frac{a_n}{\ln (1+n)}$ converges 

Note that $\displaystyle \frac{a_n}{\ln (1+n)} \leq \frac{a_n}{-\ln a_n}\iff a_n\geq \frac{1}{n+1}$
If $a_n\geq \frac{1}{n+1}$ for sufficiently many $n$, the series $\sum \frac{a_n}{\ln (1+n)}$ can thefore be compared with $\sum \frac{a_n}{-\ln a_n}$ and we're done. Otherwise, $a_n< \frac{1}{n+1}$ and we're tempted to consider the series $\displaystyle \sum \frac{1}{(n+1)\ln (1+n)}$ which is unfortunately divergent.
That's as I far as I can go with  this problem. Summation by part is a dead end as well.

Edit/Remark: this is very similar in spirit to mathguy's answer below/above; only a lot more verbose.

We will start from the "trivial" equality
$$
\frac{a_n}{\ln(n+1)}
= \frac{a_n}{\ln(n+1)} \mathbb{1}_{\left\{ a_n < \tau_n\right\}}
+ \frac{a_n}{\ln(n+1)} \mathbb{1}_{\left\{ a_n \geq \tau_n \right\}}
$$
which holds for any choice of "threshold" $(\tau_n)_n$; specifically, we want to choose a suitable sequence of thresholds which would enable us to invoke theorems of comparison. (In what you did, you implicitly chose $\tau_n = \frac{1}{n+1}$: this is not enough, so we'll choose a smaller threshold.)
Moreover, to make $\ln(n+1)$ appear somewhere (namely, when upper bounding the second term of the above identity), we will want to choose $\tau_n=\frac{1}{(n+1)^\alpha}$ for some $\alpha > 0$: so that $\ln\frac{1}{\tau_n} = \alpha\ln(n+1).$ 
(Stop now if you didn't want the full solution, but only a hint.)

In detail: As we'll see, any $\alpha > 1$ will do -- below, we set $\alpha \stackrel{\rm def}{=}2$.
Write 
$$
\frac{a_n}{\ln(n+1)}
= \frac{a_n}{\ln(n+1)} \mathbb{1}_{\left\{ a_n < \frac{1}{(n+1)^2} \right\}}
+ \frac{a_n}{\ln(n+1)} \mathbb{1}_{\left\{ a_n \geq \frac{1}{(n+1)^2} \right\}}
$$
For the first term, we can write $$\frac{a_n}{\ln(n+1)} \mathbb{1}_{\left\{ a_n < \frac{1}{(n+1)^2} \right\}}\leq \frac{1}{(n+1)^2\ln(n+1)} \mathbb{1}_{\left\{ a_n < \frac{1}{(n+1)^2} \right\}} \leq \frac{1}{(n+1)^2\ln(n+1)}$$
For the second,
$$\begin{align}
\frac{a_n}{\ln(n+1)} \mathbb{1}_{\left\{ a_n \geq \frac{1}{(n+1)^2} \right\}}
&=
\frac{a_n}{\frac{1}{2}\ln((n+1)^2)} \mathbb{1}_{\left\{ a_n \geq \frac{1}{(n+1)^2} \right\}}
\leq \frac{a_n}{\frac{1}{2}\ln \frac{1}{a_n}} \mathbb{1}_{\left\{ a_n \geq \frac{1}{(n+1)^2} \right\}} \\
&\leq \frac{2a_n}{\ln \frac{1}{a_n}} \mathbb{1}_{\left\{ a_n \geq \frac{1}{(n+1)^2} \right\}}  
\leq \frac{2a_n}{\ln \frac{1}{a_n}}. 
\end{align}$$
Together, we get
$$
0 < \frac{a_n}{\ln(n+1)} \leq \frac{1}{(n+1)^2\ln(n+1)} + 2\frac{a_n}{\ln \frac{1}{a_n}}
$$
which allows you to conclude by theorems of comparison.
