Concur with eatmeimadanish. Conda first, then pip makes the most sense given your *conda starting point.
The TL;DR Backstory
Anaconda (the distribution) and Conda (the package manager) were designed to solve installation and integration problems that the status quo did not.
The status quo here covers enormous ground: whatever combination of Python binaries (either OS-provided or downloaded from Python.org), system level package installers (e.g. apt get, yum, homebrew), Python-focused package installers (e.g. easy_install and pip), and setup frameworks (e.g. setuptools and distutils) you might happen to be using. And this status quo has evolved mightily over the years, with some parts (e.g. easy_install, distutils) falling away, and new parts (e.g. wheels, twine) coming onboard. It's not seen the vast and persistent flux of the JavaScript ecosystem, but Python packaging and install has never really been "a solved problem," and the preferred contestants for solving the problem(s) have varied greatly over time. You could argue that some or most of the problems that the native Python tools used to have are now basically solved. At least some are, but the *conda community would very much disagree that they've been eclipsed.
If you are starting from Anaconda or miniconda, I assume you enjoy their virtues (their arguably higher ease of installation, better integration, etc.). Otherwise you'd probably choose a more "stock" or vanilla Python distribution, or maybe a different "better than the base Python because X, Y, and Z" distribution (e.g. ActivePython, Enthought Canopy, ...). Given that, I'd think you'd want to use conda first, falling back to pip, rather than the other way around.
You can of course try installing each package with pip and fall back to conda only if pip disappoints, but that seems to circumvent your primary choice of starting with and favoring the *conda tools and ecosystem.