Employment Volatility and the Relevance of Aggregate Wage Rigidity The Mortensen-Pissarides (MP) framework of search labor is the thing in analyzing (equilibrium) unemployment over the business cycle. Shimer showed that the lack of hires is the big margin that affects the volatility of labor. Since then people tried to analyze potential wage rigidity for new hires.

Is it safe to conclude that - in this context - wage rigidity of old matches does not matter anymore? Is this somewhat depending on the framework (As in: old wages don't matter for hires in MP because of XYZ, but this is a strong assumption)?

Is wage rigidity, conditional on the match being new, the only relevant metric onwards?
Wage rigidity is not the only solution to the Shimer Puzzle (Shimer, AER 2005) that has been proposed in the literature, so I wouldn't say that it is the only relevant metric to consider when thinking about labor market dynamics in this context. However, as I understand the question, it aims at the role of wage rigidity in a Diamond-Mortensen-Pissarides (DMP) model and not on alternative solutions to the Shimer Puzzle.

Thus, here are a few thoughts on the rigidity of wages of newly hired workers:

The baseline DMP model is essentially a model of new hires. The wage formation mechanism as it is described in Pissarides (2000) does not differentiate between newly hired workers and ongoing employment relationships. The Nash bargaining solution dictates a surplus sharing rule that splits up what is being produced in every period, no matter how long the employment spell lasts. The bargaining powers are exogenous. In the baseline model, the elasticity of wages with respect to labor productivity is simply 1. Wages are fully flexible and immediately adjust to changes in labor productivity. This model feature lies at the heart of the Shimer puzzle. The model's inability to generate dynamics consistent with volatilities we observe in the data is due to the fact that wages adjust upwards too much in response to a positive shock, thereby lowering incentives of firms to post additional vacancies.

Since the theoretical model treats every match in every period like a new match, the rigidity of new matches is of primary empirical interest for the calibration and analysis of this class of models. 

Haefke et al. (JME, 2013) report empirical evidence for wage rigidity using US labor market time series data.  They find that the wage of new hires (both out of unemployment and after a job-to-job switch) corresponds almost one-to-one to changes in labor productivity. The estimated elasticity is around 0.8 (with quite a large standard error of 0.4). The wage of workers in ongoing employment relationships, on the other hand, shows little flexibility. The authors note that the elasticity of 0.8 is an appropriate calibration target for search and matching models. 

Whether focusing on new hires only is too strong an assumption depends mainly on the country one looks at. The US labor market, for example, is driven by huge turnovers. Thus, in every period, a large share of employees can be considered new hires. For some European labor markets, on the other hand, the assumption might be less reasonable because average employment spells tend to be longer and labor market flows smaller. 

References:

Haefke, Christian; Sonntag, Marcus; Rens, Thijs van: Wage rigidity and job creation. In: Journal of Monetary Economics 60 (2013), No. 8, pp. 887-899

Pissarides, Christopher A.: Equilibrium Unemployment Theory. 2nd Edition. The MIT Press, 2000

Shimer, Robert: The Cyclical Behavior of Equilibrium Unemployment and Vacancies. In: The American Economic Review 95 (2005), No. 1, pp. 25-49