The evolution from Pre-Proto-Indo-European (the language reconstructed by internal reconstruction from Proto-Indo-European, the language reconstructed by the comparative method from the Indo-European languages) to Old Irish is a good illustrative example (with illustration along the way from Greek and Latin) of one way to do this.
Pre-Proto-Indo-European has two conjugations, with each verb belonging to just one of the two classes: eventive (which has both active and mediopassive endings) and stative (which has no voice distinctions).
Early on, the eventive verbs develop a set of imperative endings through unclear means, but they are clearly closely related to the previous "indicative" endings.
At some point, lexical aspect (aktionsart) develops among the eventive verb, with some verbs being felt to be imperfective, and others perfective.
The imperfective eventive verbs then start being marked with the hic-et-nunc "here and now" particle (-i in the active and -r in the mediopassive) when describing present tense events.
We have now arrived at the situation we see in Proto-Indo-European. Imperfective verbs have two sets of endings, a "primary" set for present tense events, and a "secondary" set shared with perfective verbs for past tense events.
Now we start to interpret the previously lexical aspect distinction as inflectional with what was previously various methods of deriving verbs of one lexical aspect from another as ways of inflecting a single verb.
The imperfective verb gives the present and imperfect "tense" (both with "present" or imperfective aspect), the perfective verb gives the aorist, and the old stative is reinterpreted as a perfect.
At this point, other derived verbs become grammaticalised giving subjunctive, optative, and future "tenses". The original sense of the subjunctive and optative is unclear, and these were likely grammaticalised at a very early stage (at the latest only shortly after Proto-Indo-European itself), but the future is derived from what was previously a means of forming desiderative verbs (i.e. verbs with the sense "to want to").
Now we have the following:
- Present (active & mediopassive voices; indicative, subjunctive, optative, and imperative moods)
- Future (active & mediopassive voices; indicative, subjunctive, optative, and imperative moods)
- Imperfect (active & mediopassive voices)
- Aorist (active & mediopassive voices; indicative, subjunctive, optative, and imperative moods)
- Perfect (indicative, subjunctive, optative, and imperative moods)
By applying aorist endings to the perfect stem, we can then form a pluperfect (as in Greek & Latin). By applying future endings to it we can form a future perfect (as in Latin), and by applying imperfect endings to the future stem we can form a conditional (as in Old Irish).
Around this stage, Greek produces a genuine passive (as opposed to mediopassives) in certain tenses with a new suffix, possibly derived from a reflex of *dʰeh₁- "to do". Greek also introduces the augment from an earlier particle, explicitly marking past tense verbs as such (essentially the opposite of the earlier hic-et-nunc particle).
Meanwhile Italic & Celtic reinterpret the mediopassive as a true passive (with the deponents left behind as a relic of the earlier sense). Italic then replaces several of these forms with new ones formed by suffixing the stem with a reflex of *bʰuH- "to become".
Italic & Celtic also then merge the perfect and aorist, with some verbs (all of them in Italic) taking a new set of endings formed from combining both sets of endings.
Then, to get to Old Irish it's a process of accreting as many clitics as you can:
- Conjunction
- Relative Particle
- Negative Particles
- Object Pronouns
- Perspectivity/Augmentation (from a prepositional preverb but now giving either a retrospective/perfect or potential sense depending on the tense of the verb)
- Lexical preverbs (often without well defined semantics themselves)
- Subject pronoun (the nota augens)
At this point you have a verbal complex containing up to 12 different elements which can occur in many (but not all) combinations with many elements having little in the way of obvious semantics of their own, but only really making sense when the entire complex is considered as one.
It starts looking a lot closer to Georgian than the more typically Indo-European Greek and Latin seen earlier did, and much more so than the Pre-Proto-Indo-European we started with. In fact, David Stifter, a specialist in Old Irish has gone so far as to describe it as polysynthetic (although in doing so he does use a broader definition than is typical).
I'd recommend reading his grammar of Old Irish (or his chapter in the Celtic Languages volume from the Cambridge Languages Survey) for a fuller account of Old Irish and its derivation from earlier Proto-Celtic, as well as Willi's Origins of the Greek Verb for an account of the development from Pre-Proto-Indo-European up to the typical Indo-European level of Greek and Sanskrit (but also Proto-Celtic).
The key point here is that you've got a lot of time. You don't need to derive everything in one go, but can instead do so in many stages, and even use essentially the same process multiple times (note the use in Greek of both the hic-et-nunc particle and the augment), especially if the sense of the earlier process has been lost.